Berenstein–Zalcman Pizzetti Formulas for the Rank-One Opdam–Cherednik Laplacian

Abstract

We develop a Pizzetti calculus for the rank-one Opdam–Cherednik setting based on the Weyl symmetrized generalized translation. Its spectral multiplier is the Jacobi function, and the Jacobi differential equation yields a Darboux transmutation between the radial Jacobi operator and the shifted Opdam–Cherednik Laplacian. The classical hypergeometric expansion of the Jacobi multiplier is thereby converted into a Pizzetti series that converges to the symmetrized mean in the Opdam–Cherednik Schwartz topology on each compact spectral band, with explicit shifted factors of Berenstein–Zalcman type. The convergence of the series and the termwise operations are justified using the Opdam–Cherednik Schwartz and Paley–Wiener theories. We also obtain mean-value and finite shifted-polyharmonic characterizations, a normalized ball-mean expansion, and a quantitative two-radius estimate. The even specialization recovers the known radial Laplace–Beltrami calculus on rank-one noncompact symmetric spaces, while a precise curvature contraction yields the rank-one rational Dunkl Pizzetti formula. The aspect specific to the present formulation is that these identities are established for a reflection–differential spatial operator before restriction to the even sector; no novelty is claimed for the underlying Jacobi expansion.

1. Introduction

Pizzetti formulas connect geometric averaging with differential operators. For the Euclidean spherical mean

$$M_r^{\mathrm{E}}f\left(x\right)={\displaystyle \frac{1}{|{\mathbb{S}}^{n-1}|}}{\int }_{{\mathbb{S}}^{n-1}}f(x+r\omega )\mathrm{d}\omega ,$$

a real-analytic function satisfies

$$M_r^{\mathrm{E}}f\left(x\right)=\mathrm{\Gamma }\left({\displaystyle \frac{n}{2}}\right){\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(r/2)}^{2m}}{m!\mathrm{\Gamma }(\frac{n}{2}+m)}}{\mathrm{\Delta }}^{m}f\left(x\right).$$

(1)

Thus, spherical averaging is locally encoded by the Laplacian. Since Pizzetti’s original work [1], this principle has been closely related to mean-value properties, polyharmonicity, integral geometry, and the Pompeiu problem.

On curved spaces, pure powers of the Laplacian are replaced by curvature-dependent shifted products. Berenstein and Zalcman [2,3] showed that on spaces of constant curvature and on symmetric spaces these factors are generated by the radial Darboux equation for spherical means. The shifts, therefore, reflect the radial geometry itself. The purpose of the present paper is to identify and develop the analogous mechanism in rank-one Opdam–Cherednik analysis, where differential operators are coupled to reflection and the spherical spectral theory is replaced by the nonsymmetric Opdam–Cherednik transform.

Let

$$\alpha \ge \beta \ge -{\displaystyle \frac{1}{2}},\alpha >-{\displaystyle \frac{1}{2}},\rho =\alpha +\beta +1,$$

and let $Rf\left(x\right)=f(-x)$. We consider the rank-one Opdam–Cherednik operator

$$T_{\alpha ,\beta }={\displaystyle \frac{d}{dx}}+{\displaystyle \frac{1}{2}}\left((2\alpha +1)cothx+(2\beta +1)tanhx\right)(I-R)-\rho R,$$

(2)

and its shifted square

$${\mathcal{L}}_{\alpha ,\beta }=T_{\alpha ,\beta }^2-{\rho }^{2}I.$$

(3)

The shift is chosen so that, on even functions, the operator reduces to the classical Jacobi differential operator

$${\left.{\mathcal{L}}_{\alpha ,\beta }\right|}_{\mathrm{even}}={\displaystyle \frac{d^2}{d{x}^2}}+\left((2\alpha +1)cothx+(2\beta +1)tanhx\right){\displaystyle \frac{d}{dx}}.$$

(4)

The nonsymmetric spectral resolution is governed by the Opdam hypergeometric functions $G_{\lambda }^{(\alpha ,\beta )}$, while their even parts are the Jacobi functions ${\phi }_{\lambda }^{(\alpha ,\beta )}$; see [4,5,6].

The product formula of Anker, Ayadi and Sifi [7] yields generalized translations ${\tau }_r^{\alpha ,\beta }$ on the real line. The central observation of this paper is that the symmetrized translation

$${\mathcal{M}}_r^{\alpha ,\beta }f={\displaystyle \frac{1}{2}}\left({\tau }_r^{\alpha ,\beta }f+{\tau }_{-r}^{\alpha ,\beta }f\right)$$

(5)

has the Jacobi multiplier

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

(6)

Symmetrization, therefore, removes the odd part of the nonsymmetric Opdam multiplier while retaining the full differential–reflection operator in the space variable. The Jacobi differential equation then becomes the Darboux transmutation

$$\begin{array}{cc}\hfill & \left[{\partial }_r^2+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\partial }_r\right]U(r,x)\hfill \\ \hfill & ={\mathcal{L}}_{\alpha ,\beta ,x}U(r,x),U(r,x)={\mathcal{M}}_r^{\alpha ,\beta }f\left(x\right),\hfill \end{array}$$

(7)

with the regular data $U(0,x)=f\left(x\right)$ and ${\partial }_{r}U(0,x)=0$. This transmutation is the analytic source of the Pizzetti factors.

After the change of variable $t={sinh}^{2}r$, the coefficient recurrence associated with (7) produces

$${\mathcal{P}}_0^{\alpha ,\beta }\left(z\right)=1,{\mathcal{P}}_m^{\alpha ,\beta }\left(z\right)={\displaystyle \prod _{j=0}^{m-1}}\left[z-4j(j+\rho )\right],m\ge 1,$$

(8)

and hence the Berenstein–Zalcman Pizzetti expansion

$${\mathcal{M}}_r^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{1}{m!{(\alpha +1)}_m}}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f.$$

(9)

A principal analytic point is that this series is proved as a convergent identity on compact Opdam–Cherednik spectral bands rather than used formally. The proof combines the multiplier formula with the Schwartz-space isomorphism, Paley–Wiener theory, and spectral estimates for the Opdam hypergeometric kernel.

Here “compact spectral band” means that both ${\mathcal{F}}_{\alpha ,\beta }f$ and ${\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)$ are supported in a fixed interval $[-R,R]$, as made precise in (71). The expansion is also local in the radius variable; the hypergeometric power series is used for

$$|sinhr|<1.$$

This restriction concerns the Pizzetti series, not the existence of the generalized mean itself, which is defined for all real r on the underlying translation class.

We emphasize the precise scope of the novelty. The hypergeometric series of the Jacobi function in the variable $-{sinh}^{2}r$ is classical. The contribution is to place that scalar expansion inside the nonsymmetric Opdam–Cherednik calculus. We construct the Weyl-symmetrized mean, prove its Darboux transmutation in the full Schwartz topology, identify the resulting shifted products as polynomials of the reflection–differential operator ${\mathcal{L}}_{\alpha ,\beta }$, and justify the resulting Pizzetti series as a convergent identity in the compact spectral-band Schwartz topology. Thus, the nontrivial point is not the formal Taylor expansion itself, but its operator-theoretic realization before passage to the even sector. The curvature contraction and the local mean-value results are then formulated and proved in this same nonsymmetric framework.

The terminology “Berenstein–Zalcman” is used here specifically for the shifted product factorization. We do not claim to reproduce or extend in this paper the deeper spectral synthesis and Pompeiu results of [2,3]. Likewise, the even symmetric space specialization is included as a consistency theorem, not as a new construction of the classical Jacobi or Laplace–Beltrami theory, and the two-radius estimate below is presented as a quantitative Opdam–Cherednik counterpart of a classical setting rather than as a new general Pompeiu theorem.

For clarity, the ingredients entering the construction have different status. The Opdam–Cherednik product formula, the Jacobi function and its hypergeometric expansion, the classical Darboux philosophy for spherical means, and the Berenstein–Zalcman shifted-product mechanism belong to the existing theory. We do not claim novelty for these ingredients, nor for Weyl symmetrization as an abstract operation. The contribution claimed here is deliberately narrower: these classical ingredients are assembled in the rank-one nonsymmetric Opdam–Cherednik setting so that the symmetrized translation has the Jacobi multiplier while the spatial operator remains the full reflection–differential operator. Within that framework we justify the resulting Pizzetti series as a convergent operator identity in the two-sided compact spectral-band Schwartz topology and derive the stated local consequences. Thus, the novelty claim concerns this operator-theoretic realization and its rigorous convergence in the nonsymmetric setting, not the underlying scalar Jacobi identities, the classical symmetric-space theory, or the general Berenstein–Zalcman principle.

The applications retained in the paper are internal to harmonic analysis. The Pizzetti coefficients yield a mean-value characterization of ${\mathcal{L}}_{\alpha ,\beta }$-harmonic functions, a finite shifted-polyharmonic principle, and a normalized ball-mean expansion. The multiplier representation also gives a quantitative two-radius stability estimate. The geometric scope is made explicit by identifying the even Opdam–Cherednik Laplacian with the radial Laplace–Beltrami operator on rank-one noncompact Riemannian symmetric spaces. This includes the real, complex, quaternionic, and Cayley hyperbolic families; these specializations are used as consistency checks with the classical theory. A curvature scaling then contracts the nonsymmetric theory to the rank-one rational Dunkl setting and its Bessel-type Pizzetti expansion. For completeness, a short final remark records that the finite Pizzetti truncations can also be read as reflection-equivariant polynomial filters.

The organization is as follows. Section 2 collects the rank-one Opdam–Cherednik spectral framework, and Section 3 develops generalized translations, the symmetrized mean, and the Darboux equation. Section 4 proves the Pizzetti expansion. Section 5 treats curvature scaling and the rational Dunkl limit, while Section 6 discusses the real-hyperbolic and general rank-one symmetric-space specializations together with the two-radius estimate. Section 7.1 gives the mean-value, shifted-polyharmonic, and ball-mean consequences, and Section 7.2 contains a brief spectral-filter interpretation. Section 8 concludes the paper.

2. The Rank-One Opdam–Cherednik Framework

Throughout the paper we assume

$$\alpha \ge \beta \ge -{\displaystyle \frac{1}{2}},\alpha >-{\displaystyle \frac{1}{2}},\rho =\alpha +\beta +1.$$

(10)

We denote by R the reflection operator

$$\left(Rf\right)\left(x\right)=f(-x),x\in \mathbb{R},$$

and use the Jacobi weight

$$A_{\alpha ,\beta }\left(x\right)={|sinhx|}^{2\alpha +1}{(coshx)}^{2\beta +1},x\in \mathbb{R}.$$

(11)

The first-order differential–reflection operator used below is often called the Jacobi–Cherednik operator in the rank-one literature. Since its normalized eigenfunctions are the nonsymmetric Opdam hypergeometric functions and its Fourier analysis is the Opdam–Cherednik transform, we use the term rank-one Opdam–Cherednik operator for $T_{\alpha ,\beta }$ throughout the paper. We reserve “Opdam hypergeometric function” for $G_{\lambda }^{(\alpha ,\beta )}$, “Jacobi function” for ${\phi }_{\lambda }^{(\alpha ,\beta )}$, and “Opdam–Cherednik transform” for ${\mathcal{F}}_{\alpha ,\beta }$. The alternative name “Jacobi–Cherednik operator” is mentioned only to identify terminology used in part of the literature. The standing assumptions (10) remain in force unless a specialization is stated explicitly. The rank-one Opdam–Cherednik operator is the differential–reflection operator

$$\begin{array}{cc}\hfill T_{\alpha ,\beta }f\left(x\right)=& f^{\prime }\left(x\right)+{\displaystyle \frac{1}{2}}\left((2\alpha +1)cothx+(2\beta +1)tanhx\right)\left(f\left(x\right)-f(-x)\right)\hfill \\ \hfill & -\rho f(-x),x\ne 0.\hfill \end{array}$$

(12)

This is the normalization used in [5,7]; in the notation of [8] one has $k=\alpha +{\displaystyle \frac{1}{2}}$, $k^{\prime }=\beta +{\displaystyle \frac{1}{2}}$, and $\rho =k+k^{\prime }$. The second-order operator used throughout this paper is the shifted square

$${\mathcal{L}}_{\alpha ,\beta }=T_{\alpha ,\beta }^2-{\rho }^{2}I.$$

(13)

A substantial part of the Opdam–Cherednik literature denotes the unshifted operator $T_{\alpha ,\beta }^2$ by the generalized Laplacian. Its transform symbol is $-{\lambda }^2$. Our shifted operator (13), therefore, has symbol $-({\lambda }^2+{\rho }^2)$, which is the normalization compatible with the Jacobi and Laplace–Beltrami eigenvalue equations used below. A direct calculation from (12) gives

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\alpha ,\beta }f\left(x\right)=& f^{″}\left(x\right)+\left((2\alpha +1)cothx+(2\beta +1)tanhx\right)f^{\prime }\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left(-(2\alpha +1){csch}^{2}x+(2\beta +1){sech}^{2}x\right)\left(f\left(x\right)-f(-x)\right).\hfill \end{array}$$

(14)

The apparent singularity at $x=0$ in (14) is removable. For $f\in C^{\infty }\left(\mathbb{R}\right)$ we henceforth define the value at the origin by

$${\mathcal{L}}_{\alpha ,\beta }f\left(0\right)=2(\alpha +1)f^{″}\left(0\right).$$

(15)

Indeed, the Taylor expansions of $cothx$, ${csch}^{2}x$, and $f\left(x\right)-f(-x)$ show that the two terms proportional to $f^{\prime }\left(0\right)/x$ cancel, while the remaining constant term is $[1+(2\alpha +1)]f^{″}\left(0\right)$. Thus, (14), completed by (15), defines a smooth function and can be iterated on $C^{\infty }\left(\mathbb{R}\right)$ without ambiguity. In particular,

$$R{\mathcal{L}}_{\alpha ,\beta }={\mathcal{L}}_{\alpha ,\beta }R,$$

(16)

so ${\mathcal{L}}_{\alpha ,\beta }$ preserves parity. If f is even, the reflection term in (14) disappears and

$${\mathcal{L}}_{\alpha ,\beta }f=f^{″}+\left((2\alpha +1)cothx+(2\beta +1)tanhx\right)f^{\prime },Rf=f.$$

(17)

Thus, the even restriction is precisely the classical Jacobi differential operator. The local singular structure of (14) also makes the rational Dunkl limit transparent.

We recall the notation for the Gauss hypergeometric function used throughout the sequel. For $\left|z\right|<1$,

$${}_{2}F_1(a,b;c;z)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\left(a\right)}_m{\left(b\right)}_m}{{\left(c\right)}_{m}m!}}{z}^m,$$

(18)

provided $c\notin \{0,-1,-2,\dots \}$. Here ${\left(a\right)}_m$ denotes the Pochhammer symbol,

$${\left(a\right)}_0=1,{\left(a\right)}_m=a(a+1)\dots (a+m-1)={\displaystyle \frac{\mathrm{\Gamma }(a+m)}{\mathrm{\Gamma }\left(a\right)}},m\ge 1,$$

(19)

whenever the gamma quotient is defined. These conventions will be used repeatedly in the spectral factorization and in the Pizzetti coefficients.

The symmetric eigenfunctions are the Jacobi functions

$${\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right)={}_{2}F_1\left({\displaystyle \frac{\rho +i\lambda }{2}},{\displaystyle \frac{\rho -i\lambda }{2}};\alpha +1;-{sinh}^{2}x\right),\lambda \in \mathbb{C}.$$

(20)

They are even in x and are characterized by the regular spectral problem

$$\left\{\begin{array}{c}{\mathcal{L}}_{\alpha ,\beta }y=-({\lambda }^2+{\rho }^2)y,\hfill \\ y\left(0\right)=1,y^{\prime }\left(0\right)=0.\hfill \end{array}\right.$$

(21)

The corresponding nonsymmetric Opdam hypergeometric function is

$$G_{\lambda }^{(\alpha ,\beta )}\left(x\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right)-{\displaystyle \frac{1}{\rho -i\lambda }}{\displaystyle \frac{d}{dx}}{\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right),$$

(22)

or, equivalently,

$$G_{\lambda }^{(\alpha ,\beta )}\left(x\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right)+{\displaystyle \frac{\rho +i\lambda }{4(\alpha +1)}}sinh\left(2x\right){\phi }_{\lambda }^{(\alpha +1,\beta +1)}\left(x\right).$$

(23)

It is the unique solution of

$$\left\{\begin{array}{c}{T}_{\alpha ,\beta }{G}_{\lambda }^{(\alpha ,\beta )}=i\lambda G_{\lambda }^{(\alpha ,\beta )},\hfill \\ G_{\lambda }^{(\alpha ,\beta )}\left(0\right)=1.\hfill \end{array}\right.$$

(24)

Its even part is the Jacobi function:

$${\displaystyle \frac{1}{2}}\left(G_{\lambda }^{(\alpha ,\beta )}\left(x\right)+G_{\lambda }^{(\alpha ,\beta )}(-x)\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right).$$

(25)

Combining (13) with (24) gives

$${\mathcal{L}}_{\alpha ,\beta }{G}_{\lambda }^{(\alpha ,\beta )}=-({\lambda }^2+{\rho }^2)G_{\lambda }^{(\alpha ,\beta )}.$$

(26)

For later use we record the estimate needed for the Schwartz-valued differentiations in Section 3. In the parameter correspondence

$$k=\alpha +{\displaystyle \frac{1}{2}},k^{\prime }=\beta +{\displaystyle \frac{1}{2}},\rho =k+k^{\prime },$$

the rank-one Opdam–Cherednik estimate of [8] (Proposition 2.2) gives the following. If p and q are polynomials of degrees m and n, respectively, then there exists $C_{p,q}>0$ such that

$$\left|p\left({\displaystyle \frac{\partial }{\partial \lambda }}\right)q\left({\displaystyle \frac{\partial }{\partial x}}\right)G_{\lambda }^{(\alpha ,\beta )}\left(x\right)\right|\le C_{p,q}{(1+|x\left|\right)}^n{(1+|\lambda \left|\right)}^{m}exp\left(-\rho \left|x\right|+|Im\lambda |\left|x\right|\right),$$

(27)

for $x\in \mathbb{R}$ and $\lambda \in \mathbb{C}$. In particular, for every compact interval $J\subset \mathbb{R}$ and $a,b\in {\mathbb{N}}_0$, there exist $C_{J,a,b}>0$ and $N_{a,b}\in {\mathbb{N}}_0$ such that

$$\underset{x\in J}{sup}\left|{\partial }_x^a{\partial }_{\lambda }^b{G}_{\lambda }^{(\alpha ,\beta )}\left(x\right)\right|\le C_{J,a,b}{(1+|\lambda \left|\right)}^{N_{a,b}},\lambda \in \mathbb{R}.$$

(28)

By the symmetrization identity (25), the same local polynomial-growth estimate holds for ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(x\right)$.

For

$$f\in L^1\left(\mathbb{R},A_{\alpha ,\beta }\left(x\right)\mathrm{d}x\right),$$

the Opdam–Cherednik transform is defined by

$${\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)={\int }_{\mathbb{R}}f\left(x\right)G_{\lambda }^{(\alpha ,\beta )}(-x)A_{\alpha ,\beta }\left(x\right)\mathrm{d}x,\lambda \in \mathbb{R}.$$

(29)

In the present normalization one has the $L^1$–$L^{\infty }$ estimate

$$\parallel {\mathcal{F}}_{\alpha ,\beta }{f\parallel }_{L^{\infty }\left(\mathbb{R}\right)}\le 2{\parallel f\parallel }_{L^1(\mathbb{R},A_{\alpha ,\beta }\left(x\right)\mathrm{d}x)},$$

(30)

see [8]. If $f\in C_c^{\infty }\left(\mathbb{R}\right)$, then (29) extends to an entire function of $\lambda$. The Harish–Chandra c-function is

$$c_{\alpha ,\beta }\left(\lambda \right)={\displaystyle \frac{2^{\rho -i\lambda }\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }\left(i\lambda \right)}{\mathrm{\Gamma }\left(\frac{\rho +i\lambda }{2}\right)\mathrm{\Gamma }\left(\frac{\alpha -\beta +1+i\lambda }{2}\right)}}.$$

(31)

For suitable h, the inverse transform is

$${\mathcal{F}}_{\alpha ,\beta }^{-1}h\left(x\right)={\int }_{\mathbb{R}}h\left(\lambda \right)G_{\lambda }^{(\alpha ,\beta )}\left(x\right)\left(1-{\displaystyle \frac{\rho }{i\lambda }}\right){\displaystyle \frac{\mathrm{d}\lambda }{8\pi |c_{\alpha ,\beta }{\left(\lambda \right)|}^2}},$$

(32)

where the integrand at $\lambda =0$ is understood by its regular limiting value. The associated Parseval identity is

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}f\left(x\right)g(-x)A_{\alpha ,\beta }\left(x\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right){\mathcal{F}}_{\alpha ,\beta }g(-\lambda )\left(1-{\displaystyle \frac{\rho }{i\lambda }}\right){\displaystyle \frac{\mathrm{d}\lambda }{8\pi |c_{\alpha ,\beta }{\left(\lambda \right)|}^2}},\hfill \end{array}$$

(33)

for Schwartz functions f and g. Opdam’s positive Plancherel formula is, therefore, written as

$${\parallel f\parallel }_{L_{\alpha ,\beta }^2}^2={\int }_0^{\infty }\left(|{\mathcal{F}}_{\alpha ,\beta }{f\left(\lambda \right)|}^2+{\left|{\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)\left(\lambda \right)\right|}^2\right){\displaystyle \frac{\mathrm{d}\lambda }{16\pi |c_{\alpha ,\beta }{\left(\lambda \right)|}^2}}.$$

(34)

where

$$L_{\alpha ,\beta }^2\left(\mathbb{R}\right)=L^2\left(\mathbb{R},A_{\alpha ,\beta }\left(x\right)\mathrm{d}x\right).$$

(35)

We use the shorthand $L_{\alpha ,\beta }^2\left(\mathbb{R}\right)$ consistently below for this weighted Hilbert space. See [5,7,8] for these transform identities and their equivalent normalizations.

The natural Schwartz space is

$${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)={(coshx)}^{-\rho }\mathcal{S}\left(\mathbb{R}\right),$$

(36)

endowed with the Fréchet topology transported from $\mathcal{S}\left(\mathbb{R}\right)$ by the map $f\mapsto {(coshx)}^{\rho }f\left(x\right)$. The Opdam–Cherednik transform and its inverse are topological isomorphisms

$${\mathcal{F}}_{\alpha ,\beta }:{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)⟶\mathcal{S}\left(\mathbb{R}\right).$$

(37)

This is the rank-one Schwartz theorem; see [7,8,9]. In particular, ${\mathcal{F}}_{\alpha ,\beta }$ is injective on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$.

The corresponding Paley–Wiener theorem may be stated in support-radius form. If $R>0$ and

$$f\in C_c^{\infty }\left(\mathbb{R}\right),suppf\subseteq [-R,R],$$

then ${\mathcal{F}}_{\alpha ,\beta }f$ extends to an entire function and, for every $N\in \mathbb{N}$, there exists $C_N>0$ such that

$$|{\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)|\le C_N{(1+|\lambda \left|\right)}^{-N}{e}^{R|Im\lambda |},\lambda \in \mathbb{C}.$$

(38)

Conversely, an entire function satisfying (38) for all N is the Opdam–Cherednik transform of a function in $C_c^{\infty }\left(\mathbb{R}\right)$ supported in $[-R,R]$; see [5,8]. Finally, on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$ the first- and second-order operators are diagonalized by

$${\mathcal{F}}_{\alpha ,\beta }\left(T_{\alpha ,\beta }f\right)\left(\lambda \right)=i\lambda {\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)$$

(39)

and

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }f\right)\left(\lambda \right)=-({\lambda }^2+{\rho }^2){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

(40)

Thus, $T_{\alpha ,\beta }$ and ${\mathcal{L}}_{\alpha ,\beta }$ act continuously on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$, and (27) supplies the multiplier bounds needed to justify parameter differentiation in the generalized translations and means.

3. Generalized Translation, the Symmetrized Mean, and the Darboux Equation

The product formula of Anker, Ayadi and Sifi [7] states that, for $x,y\in \mathbb{R}$ and $\lambda \in \mathbb{C}$,

$$G_{\lambda }^{(\alpha ,\beta )}\left(x\right)G_{\lambda }^{(\alpha ,\beta )}\left(y\right)={\int }_{\mathbb{R}}{G}_{\lambda }^{(\alpha ,\beta )}\left(z\right)\mathrm{d}{\mu }_{x,y}^{(\alpha ,\beta )}\left(z\right),$$

(41)

where ${\mu }_{x,y}^{(\alpha ,\beta )}$ is a compactly supported real measure which need not be positive. The measures are symmetric in x and y, have uniformly bounded total variation,

$$\underset{x,y\in \mathbb{R}}{sup}{\parallel {\mu }_{x,y}^{(\alpha ,\beta )}\parallel }_{\mathrm{TV}}<\infty ,$$

(42)

and satisfy

$$supp{\mu }_{x,y}^{(\alpha ,\beta )}\subseteq [-|x|-|y|,-|\left|x\right|-\left|y\right|\left|\right]\cup \left[\right|\left|x\right|-\left|y\right||,|x|+|y\left|\right].$$

(43)

Moreover,

$${\mu }_{0,y}^{(\alpha ,\beta )}={\delta }_y,{\mu }_{x,0}^{(\alpha ,\beta )}={\delta }_x.$$

(44)

Since $G_{i\rho }^{(\alpha ,\beta )}\equiv 1$, substitution $\lambda =i\rho$ in (41) also gives the normalization

$${\mu }_{x,y}^{(\alpha ,\beta )}\left(\mathbb{R}\right)=1.$$

(45)

For a bounded Borel function f, the generalized translation is, therefore, well defined by

$${\tau }_x^{\alpha ,\beta }f\left(y\right)={\int }_{\mathbb{R}}f\left(z\right)\mathrm{d}{\mu }_{x,y}^{(\alpha ,\beta )}\left(z\right).$$

(46)

Equations (42)–(44) imply

$${\tau }_0^{\alpha ,\beta }=I,{\tau }_x^{\alpha ,\beta }f\left(y\right)={\tau }_y^{\alpha ,\beta }f\left(x\right),\parallel {\tau }_x^{\alpha ,\beta }{f\parallel }_{\infty }\le C_{\alpha ,\beta }{\parallel f\parallel }_{\infty }.$$

For $f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$, the fundamental multiplier identity is

$${\mathcal{F}}_{\alpha ,\beta }\left({\tau }_x^{\alpha ,\beta }f\right)\left(\lambda \right)=G_{\lambda }^{(\alpha ,\beta )}\left(x\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

(47)

We spell out the extension to the full Schwartz class, in order to avoid any circularity. Proposition 4.4(v) of [7] proves (47) first for compactly supported continuous functions (hence for $C_c^{\infty }\left(\mathbb{R}\right)$). Fix $x\in \mathbb{R}$. By (28), multiplication by $G_{\lambda }^{(\alpha ,\beta )}\left(x\right)$ is a continuous endomorphism of $\mathcal{S}\left(\mathbb{R}\right)$. Consequently

$${\mathcal{T}}_{x}f:={\mathcal{F}}_{\alpha ,\beta }^{-1}\left(G_{\lambda }^{(\alpha ,\beta )}\left(x\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\right)$$

defines a continuous map ${\mathcal{T}}_x:{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\to {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$. Choose $f_n\in C_c^{\infty }\left(\mathbb{R}\right)$ with $f_n\to f$ in ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$. Then ${\mathcal{T}}_x{f}_n\to {\mathcal{T}}_{x}f$ in the same Fréchet topology. On the other hand, convergence in ${\mathcal{S}}_{\alpha ,\beta }$ implies uniform convergence on $\mathbb{R}$, and the uniform total-variation bound (42) gives, for every y,

$$\left|{\int }_{\mathbb{R}}(f_n-f)\left(z\right)\mathrm{d}{\mu }_{x,y}^{(\alpha ,\beta )}\left(z\right)\right|\le C_{\alpha ,\beta }{\parallel f_n-f\parallel }_{\infty }⟶0.$$

Since ${\mathcal{T}}_x{f}_n={\tau }_x^{\alpha ,\beta }{f}_n$ for $f_n\in C_c^{\infty }$, pointwise evaluation in the Schwartz topology shows ${\mathcal{T}}_{x}f={\tau }_x^{\alpha ,\beta }f$ as defined by the signed product measure. Hence

$${\tau }_x^{\alpha ,\beta }:{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)⟶{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$$

(48)

is continuous and (47) holds on the full Schwartz class. In particular, no interchange of the transform with the signed measure is being assumed for an arbitrary Schwartz function: it is justified on $C_c^{\infty }$ and then extended by continuity. The Schwartz theorem (37), together with the estimates (27)–(28) immediately give the regularity that will be used repeatedly below: for every $f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$,

$$x⟼{\tau }_x^{\alpha ,\beta }f\mathrm{belongs}\mathrm{to}{C}^{\infty }\left(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\right).$$

(49)

Indeed, after transformation the qth x-derivative is ${\partial }_x^q{G}_{\lambda }^{(\alpha ,\beta )}\left(x\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)$, and (28), the Leibniz rule, and the rapid decay of ${\mathcal{F}}_{\alpha ,\beta }f$ show locally uniformly in x that this product belongs to $\mathcal{S}\left(\mathbb{R}\right)$ with all Schwartz seminorms. For completeness, we record the explicit product kernel in the generic range

$$\alpha >\beta >-{\displaystyle \frac{1}{2}}.$$

Set

$$M_{\alpha ,\beta }={\displaystyle \frac{\mathrm{\Gamma }(\alpha +1)}{\sqrt{\pi }\mathrm{\Gamma }(\alpha -\beta )\mathrm{\Gamma }(\beta +\frac{1}{2})}}.$$

(50)

For $\chi \in [0,\pi ]$, define

$$g(x,y,z,\chi )=1-{cosh}^{2}x-{cosh}^{2}y-{cosh}^{2}z+2coshxcoshycoshzcos\chi ,$$

(51)

write $g_{+}=max\{g,0\}$, and, when $xy\ne 0$, put

$${\sigma }_{x,y,z}^{\chi }={\displaystyle \frac{coshxcoshy-coshzcos\chi }{sinhxsinhy}}.$$

(52)

For $x,y,z\ne 0$ satisfying

$$\left|\right|x|-|y\left|\right|<\left|z\right|<\left|x\right|+\left|y\right|,$$

the product kernel is

$$\begin{array}{cc}\hfill K_{\alpha ,\beta }(x,y,z)=& M_{\alpha ,\beta }{|sinhxsinhysinhz|}^{-2\alpha }{\int }_0^{\pi }{g}_{+}{(x,y,z,\chi )}^{\alpha -\beta -1}\hfill \\ \hfill & \times \left[1-{\sigma }_{x,y,z}^{\chi }+{\sigma }_{x,z,y}^{\chi }+{\sigma }_{z,y,x}^{\chi }+{\displaystyle \frac{\rho }{\beta +\frac{1}{2}}}cothxcothycothz{sin}^2\chi \right]\hfill \\ \hfill & \times {(sin\chi )}^{2\beta }\mathrm{d}\chi .\hfill \end{array}$$

(53)

It vanishes outside the region in (43), and for $xy\ne 0$,

$$\mathrm{d}{\mu }_{x,y}^{(\alpha ,\beta )}\left(z\right)=K_{\alpha ,\beta }(x,y,z)A_{\alpha ,\beta }\left(z\right)\mathrm{d}z.$$

(54)

The explicit density (53) is used only in the generic range $\alpha >\beta >-{\displaystyle \frac{1}{2}}$. At $\alpha =\beta >-{\displaystyle \frac{1}{2}}$ the separate endpoint kernel is given in Theorem 3.5 of [7]; more generally, Proposition 4.4 of that paper formulates the generalized translations throughout the standing range $\alpha \ge \beta \ge -{\displaystyle \frac{1}{2}}$, $\alpha >-{\displaystyle \frac{1}{2}}$. We, therefore, do not identify endpoint operators by an unspecified pointwise limit of (53). The reflection-equivariance statement at the endpoints will instead be deduced below from the Darboux uniqueness theorem, so no limiting topology for the kernel is required. For completeness, we make the endpoint references more explicit. When $\alpha =\beta >-{\displaystyle \frac{1}{2}}$, Theorem 3.5, formulas (3.9)–(3.10) of [7] gives the product measure ${\mu }_{x,y}^{(\alpha ,\alpha )}$ directly. Proposition 4.4(iv)–(v) of that paper then gives, throughout the standing parameter range, the translation product identity and its Opdam–Cherednik transform multiplier. In particular, the cases $\alpha =\beta$ and $\beta =-{\displaystyle \frac{1}{2}}$ are covered at the operator level without inserting the generic density (53) at a singular parameter value. The symmetrized multiplier is, therefore, obtained by averaging the two valid endpoint multipliers,

$${\displaystyle \frac{1}{2}}\left(G_{\lambda }^{(\alpha ,\beta )}\left(r\right)+G_{\lambda }^{(\alpha ,\beta )}(-r)\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right),$$

and the reflection equivariance identity follows, equivalently, from Corollary 1. Thus, no unproved pointwise limit of the generic kernel is used. These endpoint formulas are used at the operator level. In particular, reflection equivariance at $\alpha =\beta$ or $\beta =-\frac{1}{2}$ is not inferred by substituting an endpoint value into the generic density (53); it follows from the endpoint translation identities just cited together with the Darboux uniqueness argument in Corollary 1. We now symmetrize only the translation parameter. For $r\in \mathbb{R}$ define

$${\mathcal{M}}_r^{\alpha ,\beta }f={\displaystyle \frac{1}{2}}\left({\tau }_r^{\alpha ,\beta }f+{\tau }_{-r}^{\alpha ,\beta }f\right).$$

(55)

This is not an ad hoc averaging chosen only to force a desired multiplier. In rank one the Weyl group is $W=\{1,R\}$, and (55) is precisely the W-symmetrization of the translation parameter. On the transform side the same averaging extracts the W-invariant part of the nonsymmetric Opdam hypergeometric kernel,

$${\displaystyle \frac{1}{2}}\left(G_{\lambda }^{(\alpha ,\beta )}\left(r\right)+G_{\lambda }^{(\alpha ,\beta )}(-r)\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right).$$

Thus, ${\mathcal{M}}_r^{\alpha ,\beta }$ is the natural bridge from the nonsymmetric Opdam–Cherednik translation to the Jacobi radial multiplier, while the operator acting in the space variable remains the full reflection–differential operator ${\mathcal{L}}_{\alpha ,\beta }$. By construction,

$${\mathcal{M}}_{-r}^{\alpha ,\beta }={\mathcal{M}}_r^{\alpha ,\beta },$$

(56)

so in the sequel the radius may be restricted to $r\ge 0$. In the generic parameter range and for $rx\ne 0$, averaging (53) in the first variable gives

$${\mathcal{M}}_r^{\alpha ,\beta }f\left(x\right)={\int }_{I_{r,x}}f\left(z\right)K_{\alpha ,\beta }^{\mathrm{sym}}(r,x,z)A_{\alpha ,\beta }\left(z\right)\mathrm{d}z,$$

(57)

where

$$I_{r,x}=[-r-|x|,-|r-\left|x\right|\left|\right]\cup \left[\right|r-\left|x\right||,r+|x\left|\right]$$

and

$$\begin{array}{ll}{K}_{\alpha ,\beta }^{\mathrm{sym}}(r,x,z)=& M_{\alpha ,\beta }{|sinhrsinhxsinhz|}^{-2\alpha }\\ & \times {\int }_0^{\pi }{g}_{+}{(r,x,z,\chi )}^{\alpha -\beta -1}\left(1+{\sigma }_{z,x,r}^{\chi }\right){(sin\chi )}^{2\beta }\mathrm{d}\chi .\end{array}$$

(58)

Indeed,

$$K_{\alpha ,\beta }^{\mathrm{sym}}(r,x,z)={\displaystyle \frac{1}{2}}\left[K_{\alpha ,\beta }(r,x,z)+K_{\alpha ,\beta }(-r,x,z)\right].$$

(59)

Under $r\mapsto -r$, the quantities ${\sigma }_{r,x,z}^{\chi }$ and ${\sigma }_{r,z,x}^{\chi }$ change sign, the term containing $cothr$ changes sign, whereas $g(r,x,z,\chi )$ and ${\sigma }_{z,x,r}^{\chi }$ remain unchanged. The cancellation in (59), therefore, gives (58). At the degenerate points,

$${\mathcal{M}}_0^{\alpha ,\beta }f\left(x\right)=f\left(x\right),{\mathcal{M}}_r^{\alpha ,\beta }f\left(0\right)={\displaystyle \frac{f\left(r\right)+f(-r)}{2}}.$$

(60)

The crucial feature of the symmetrization is that the nonsymmetric Opdam multiplier is replaced exactly by the Jacobi multiplier.

Proposition 1.

Let $f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$. Then

$$r⟼{\mathcal{M}}_r^{\alpha ,\beta }f$$

is an even $C^{\infty }$ map from $\mathbb{R}$ into ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$ and, for every $r\in \mathbb{R}$,

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right),\lambda \in \mathbb{R}.$$

(61)

Moreover,

$${\mathcal{L}}_{\alpha ,\beta }{\mathcal{M}}_r^{\alpha ,\beta }f={\mathcal{M}}_r^{\alpha ,\beta }{\mathcal{L}}_{\alpha ,\beta }f,$$

(62)

and the symmetrized mean is reflection equivariant:

$$R{\mathcal{M}}_r^{\alpha ,\beta }f={\mathcal{M}}_r^{\alpha ,\beta }Rf.$$

(63)

Proof. 

The smoothness assertion follows immediately from (49) and the definition (55); evenness is (56). Applying (47) with translation parameters r and $-r$ gives

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(\lambda \right)& ={\displaystyle \frac{1}{2}}\left(G_{\lambda }^{(\alpha ,\beta )}\left(r\right)+G_{\lambda }^{(\alpha ,\beta )}(-r)\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\hfill \\ \hfill & ={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right),\hfill \end{array}$$

where the last equality is (25). This proves (61). Notice that (28) and the Schwartz isomorphism independently show that the right-hand side belongs to $\mathcal{S}\left(\mathbb{R}\right)$, so the multiplier identity is an identity in the full Schwartz topology, not merely pointwise in $\lambda$. For the commutation relation, (40) and (61) give

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }{\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(\lambda \right)& =-({\lambda }^2+{\rho }^2){\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right),\hfill \\ \hfill {\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_r^{\alpha ,\beta }{\mathcal{L}}_{\alpha ,\beta }f\right)\left(\lambda \right)& ={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)\left[-({\lambda }^2+{\rho }^2){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\right].\hfill \end{array}$$

The two transforms coincide, and injectivity of ${\mathcal{F}}_{\alpha ,\beta }$ on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$ yields (62). It remains to prove (63). Suppose first that $\alpha >\beta >-{\displaystyle \frac{1}{2}}$. From (58),

$$K_{\alpha ,\beta }^{\mathrm{sym}}(r,-x,-z)=K_{\alpha ,\beta }^{\mathrm{sym}}(r,x,z),$$

(64)

because $g(r,-x,-z,\chi )=g(r,x,z,\chi )$ and ${\sigma }_{-z,-x,r}^{\chi }={\sigma }_{z,x,r}^{\chi }$. Since $A_{\alpha ,\beta }$ is even, a change of variables $z=-u$ in (57) gives

$$\begin{array}{cc}\hfill \left(R{\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(x\right)& ={\mathcal{M}}_r^{\alpha ,\beta }f(-x)\hfill \\ \hfill & ={\int }_{\mathbb{R}}f(-u)K_{\alpha ,\beta }^{\mathrm{sym}}(r,-x,-u)A_{\alpha ,\beta }\left(u\right)\mathrm{d}u\hfill \\ \hfill & ={\int }_{\mathbb{R}}\left(Rf\right)\left(u\right)K_{\alpha ,\beta }^{\mathrm{sym}}(r,x,u)A_{\alpha ,\beta }\left(u\right)\mathrm{d}u\hfill \\ \hfill & ={\mathcal{M}}_r^{\alpha ,\beta }\left(Rf\right)\left(x\right).\hfill \end{array}$$

This proves (63) in the generic range $\alpha >\beta >-{\displaystyle \frac{1}{2}}$. The endpoint cases are not inferred from an unspecified kernel limit. They are completed, after the Darboux uniqueness theorem, by Corollary 1 below. □

The multiplier identity now yields the Darboux transmutation that drives the Pizzetti calculus. The Jacobi differential operator acts in the radius variable, whereas the shifted Opdam–Cherednik Laplacian acts in the space variable. The Schwartz-valued smoothness proved above justifies all parameter differentiations before pointwise evaluation.

Theorem 1.

Let $f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$ and, for $r\ge 0$, set

$$U(r,x)={\mathcal{M}}_r^{\alpha ,\beta }f\left(x\right).$$

Extend U to negative values of r by

$$U(-r,x)=U(r,x).$$

Then

$$U\in C^{\infty }\left(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\right)\subset C^{\infty }\left({\mathbb{R}}^2\right),$$

and, for every $r\ne 0$ and $x\in \mathbb{R}$,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(r,x)+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\displaystyle \frac{\partial U}{\partial r}}(r,x)\hfill \\ \hfill & ={\mathcal{L}}_{\alpha ,\beta ,x}U(r,x)={\mathcal{M}}_{\left|r\right|}^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }f\right)\left(x\right).\hfill \end{array}$$

(65)

The regular Cauchy data at the singular endpoint $r=0$ are

$$U(0,x)=f\left(x\right),{\displaystyle \frac{\partial U}{\partial r}}(0,x)=0,x\in \mathbb{R}.$$

(66)

Moreover, the singular coefficient in (65) is removable along the even solution, and

$$2(\alpha +1){\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(0,x)={\mathcal{L}}_{\alpha ,\beta }f\left(x\right).$$

(67)

Finally, U is the unique even function in

$$C^2\left(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\right)$$

satisfying (65) and the initial conditions (66).

Proof. 

By the multiplier identity of Proposition 1, for $r\ge 0$,

$${\mathcal{F}}_{\alpha ,\beta }\left(U(r,·)\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

(68)

Since the Jacobi function is even,

$${\phi }_{\lambda }^{(\alpha ,\beta )}(-r)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right),$$

the same identity holds for every $r\in \mathbb{R}$ after the even extension of U. Because

$$f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right),$$

the Schwartz isomorphism gives

$${\mathcal{F}}_{\alpha ,\beta }f\in \mathcal{S}\left(\mathbb{R}\right).$$

The standard estimates for the Opdam hypergeometric function and its derivatives imply that, for every compact interval $K\subset \mathbb{R}$, every $q\in {\mathbb{N}}_0$, and every derivative in the spectral variable, the functions

$${\partial }_r^q{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right),r\in K,$$

have at most polynomial growth in $\lambda$, uniformly for $r\in K$. Consequently,

$${\partial }_r^q{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\in \mathcal{S}\left(\mathbb{R}\right),$$

locally uniformly in r. Since ${\mathcal{F}}_{\alpha ,\beta }^{-1}$ is continuous from $\mathcal{S}\left(\mathbb{R}\right)$ onto ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$, it follows that

$$U\in C^{\infty }\left(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\right)$$

and, for every $q\in {\mathbb{N}}_0$,

$${\mathcal{F}}_{\alpha ,\beta }\left({\displaystyle \frac{{\partial }^{q}U}{\partial r^q}}(r,·)\right)\left(\lambda \right)={\displaystyle \frac{d^q}{d{r}^q}}{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

(69)

For clarity, evaluation after transformation is continuous in the topology being used. Since, ${\mathcal{F}}_{\alpha ,\beta }$ is a topological isomorphism by (37), and for each fixed $\lambda$ the functional $h\mapsto h\left(\lambda \right)$ is continuous on $\mathcal{S}\left(\mathbb{R}\right)$. Thus, every transformed identity above may be evaluated at a fixed spectral parameter without leaving the chosen Schwartz topology. The Jacobi function satisfies the differential equation

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d^2}{d{r}^2}}{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\displaystyle \frac{d}{dr}}{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)\hfill \\ \hfill & =-({\lambda }^2+{\rho }^2){\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right),r\ne 0.\hfill \end{array}$$

(70)

Applying the Opdam–Cherednik transform in the x variable to the left-hand side of (65) and using (69) and (70), we obtain

$$-({\lambda }^2+{\rho }^2){\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

On the other hand, the spectral identity

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }g\right)\left(\lambda \right)=-({\lambda }^2+{\rho }^2){\mathcal{F}}_{\alpha ,\beta }g\left(\lambda \right)$$

gives

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta ,x}U(r,·)\right)\left(\lambda \right)& =-({\lambda }^2+{\rho }^2){\mathcal{F}}_{\alpha ,\beta }\left(U(r,·)\right)\left(\lambda \right)\hfill \\ \hfill & =-({\lambda }^2+{\rho }^2){\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).\hfill \end{array}$$

Injectivity of ${\mathcal{F}}_{\alpha ,\beta }$, therefore, yields

$${\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\displaystyle \frac{\partial U}{\partial r}}={\mathcal{L}}_{\alpha ,\beta ,x}U.$$

The commutation identity

$${\mathcal{L}}_{\alpha ,\beta }{\mathcal{M}}_r^{\alpha ,\beta }={\mathcal{M}}_r^{\alpha ,\beta }{\mathcal{L}}_{\alpha ,\beta }$$

then gives the second equality in (65). At $r=0$, the normalization

$${\phi }_{\lambda }^{(\alpha ,\beta )}\left(0\right)=1$$

in (68) gives

$$U(0,x)=f\left(x\right).$$

Since U is even in r,

$${\displaystyle \frac{\partial U}{\partial r}}(0,x)=0,$$

which proves (66). We next examine the apparently singular endpoint. Since U is even and $C^2$ in r,

$${\displaystyle \frac{\partial U}{\partial r}}(r,x)=r{\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(0,x)+o\left(r\right)(r\to 0).$$

Moreover,

$$cothr={\displaystyle \frac{1}{r}}+O\left(r\right),tanhr=O\left(r\right).$$

Hence

$$\begin{array}{cc}\hfill & \underset{r\to 0}{lim}\left[{\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(r,x)+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\displaystyle \frac{\partial U}{\partial r}}(r,x)\right]\hfill \\ \hfill & =\left(1+2\alpha +1\right){\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(0,x)\hfill \\ \hfill & =2(\alpha +1){\displaystyle \frac{{\partial }^{2}U}{\partial r^2}}(0,x).\hfill \end{array}$$

Since

$$U(r,·)⟶f\mathrm{in}{\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)\mathrm{as}r\to 0,$$

continuity of ${\mathcal{L}}_{\alpha ,\beta }$ on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$ gives

$${\mathcal{L}}_{\alpha ,\beta ,x}U(r,·)⟶{\mathcal{L}}_{\alpha ,\beta }f.$$

Passing to the limit in (65) proves (67).

It remains only to prove uniqueness. Let V be another even $C^2(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta })$ solution with the same Cauchy data and put $W=V-U$. For fixed $\lambda \in \mathbb{R}$,

$$w_{\lambda }\left(r\right)={\mathcal{F}}_{\alpha ,\beta }\left(W(r,·)\right)\left(\lambda \right)$$

satisfies the homogeneous Jacobi equation

$$w_{\lambda }^{\prime \prime }+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right)w_{\lambda }^{\prime }+({\lambda }^2+{\rho }^2)w_{\lambda }=0,w_{\lambda }\left(0\right)=w_{\lambda }^{\prime }\left(0\right)=0.$$

In divergence form,

$${\left(A_{\alpha ,\beta }{w}_{\lambda }^{\prime }\right)}^{\prime }=-({\lambda }^2+{\rho }^2)A_{\alpha ,\beta }{w}_{\lambda }.$$

Choose $r_0\in (0,1)$ and constants $0<c_0\le C_0$ such that

$$c_0{r}^{2\alpha +1}\le A_{\alpha ,\beta }\left(r\right)\le C_0{r}^{2\alpha +1},0<r\le r_0.$$

Writing $q_{\lambda }={\lambda }^2+{\rho }^2$ and $M_{\lambda }\left(r\right)={sup}_{0\le s\le r}\left|w_{\lambda }\left(s\right)\right|$, integration of the divergence equation from 0 to $r\le r_0$ gives

$$\begin{array}{cc}\hfill |w_{\lambda }^{\prime }\left(r\right)|& \le |q_{\lambda }|A_{\alpha ,\beta }{\left(r\right)}^{-1}{\int }_0^r{A}_{\alpha ,\beta }\left(s\right)\left|w_{\lambda }\left(s\right)\right|\mathrm{d}s\hfill \\ & \le {\displaystyle \frac{|q_{\lambda }|C_0}{c_0(2\alpha +2)}}r{M}_{\lambda }\left(r\right)=:C_{\lambda }r{M}_{\lambda }\left(r\right).\hfill \end{array}$$

A second integration, using $w_{\lambda }\left(0\right)=0$, yields

$$M_{\lambda }\left(r\right)\le {\displaystyle \frac{C_{\lambda }{r}^2}{2}}{M}_{\lambda }\left(r\right).$$

After decreasing $r_0$ so that $C_{\lambda }{r}_0^2/2<1$, one obtains $M_{\lambda }\left(r_0\right)=0$. Hence $w_{\lambda }=0$ on $[0,r_0]$. Ordinary ODE uniqueness away from $r=0$ extends this identity to all $r\ge 0$, and evenness gives it for $r<0$. Injectivity of the Opdam–Cherednik transform then yields $W=0$. This proves uniqueness without a separate lengthy Volterra argument. □

Corollary 1.

For the full parameter range (10), the symmetrized mean is reflection equivariant on ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$; that is, (63) holds for every $r\in \mathbb{R}$.

Proof. 

The generic case was proved directly from the kernel symmetry in Proposition 1. It remains to consider an endpoint parameter pair. Put

$$V(r,·)={\mathcal{M}}_r^{\alpha ,\beta }\left(Rf\right),W(r,·)=R{\mathcal{M}}_r^{\alpha ,\beta }f.$$

By Theorem 1, V is an even $C^2(\mathbb{R};{\mathcal{S}}_{\alpha ,\beta })$ solution of the Darboux equation with initial value $Rf$. Because $R{\mathcal{L}}_{\alpha ,\beta }={\mathcal{L}}_{\alpha ,\beta }R$, the function W satisfies the same equation and the same initial data. The uniqueness part of Theorem 1, therefore, gives $V=W$. This proves (63) without passing to a limit of the endpoint kernels. □

4. Berenstein–Zalcman Pizzetti Formulas

The Darboux transmutation equation contains the full Berenstein–Zalcman shifted-product mechanism. In this section we derive the corresponding Pizzetti formulas for the rank-one Opdam–Cherednik Laplacian. The essential point is to obtain the series as a convergent identity in a natural spectral class rather than as a formal expansion. We, therefore, begin with the Paley–Wiener/Schwartz framework that justifies analyticity and termwise application of the Opdam–Cherednik Laplacian.

For $R>0$, we work first on the two-sided spectral band

$${\mathcal{B}}_R^{\alpha ,\beta }=\left\{f\in {\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right):supp{\mathcal{F}}_{\alpha ,\beta }f\cup supp{\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)\subset [-R,R]\right\},$$

(71)

equipped with the subspace Fréchet topology inherited from ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$. The two-sided formulation is convenient because the Plancherel identity contains the transforms of both f and $Rf$. By (40), (16), and Proposition 1, the space ${\mathcal{B}}_R^{\alpha ,\beta }$ is invariant under ${\mathcal{L}}_{\alpha ,\beta }$, reflection, and every ${\mathcal{M}}_r^{\alpha ,\beta }$. The Paley–Wiener theory recalled in Section 2 enters at this point. We use it together with the Schwartz isomorphism to work with compact spectral support in a topology that is stable under entire spectral multipliers. Concretely, if $h\in C_c^{\infty }\left(\mathbb{R}\right)$ is supported in $[-R,R]$ and $\Psi (\lambda ,t)$ is entire in $\lambda$ and holomorphic in t, then for t in a compact subset of its domain the product $\Psi (\lambda ,t)h\left(\lambda \right)$ remains in $C_c^{\infty }\left([-R,R]\right)$, depends holomorphically on t in that Fréchet space, and its inverse Opdam–Cherednik transform depends holomorphically on t in ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$. This is the precise Paley–Wiener/Schwartz input behind the band-limited holomorphic argument below; no formal power-series assumption is being made.

Holomorphy here is understood in the usual locally convex sense: the map is holomorphic for every continuous seminorm of the relevant Fréchet space, equivalently after composition with every continuous linear functional. The normal convergence estimates used below are uniform for each of these seminorms on compact t-subdisks.

Proposition 2.

Let $f\in {\mathcal{B}}_R^{\alpha ,\beta }$ and define

$$U(r,·)={\mathcal{M}}_r^{\alpha ,\beta }f.$$

Then U is even and real analytic in r near $r=0$. More precisely, if

$$t={sinh}^{2}r,$$

then there is a ${\mathcal{B}}_R^{\alpha ,\beta }$-valued holomorphic function $\tilde{U}\left(t\right)$ on the disk $\left|t\right|<1$ such that

$$\tilde{U}\left({sinh}^{2}r\right)={\mathcal{M}}_r^{\alpha ,\beta }f$$

for real r with $|sinhr|<1$. Consequently,

$$\tilde{U}\left(t\right)={\displaystyle \sum _{m=0}^{\infty }}{a}_m{t}^m,a_m\in {\mathcal{B}}_R^{\alpha ,\beta },a_0=f,$$

(72)

with convergence in the Fréchet topology, locally uniformly for $\left|t\right|<1$. The series may be differentiated term by term with respect to t, and ${\mathcal{L}}_{\alpha ,\beta }$ may be applied term by term.

Proof. 

By Proposition 1,

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_r^{\alpha ,\beta }f\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).$$

Introduce

$${\mathrm{\Phi }}_{\lambda }\left(t\right)={}_{2}F_1\left({\displaystyle \frac{\rho +i\lambda }{2}},{\displaystyle \frac{\rho -i\lambda }{2}};\alpha +1;-t\right),\left|t\right|<1.$$

(73)

For real r with $|sinhr|<1$,

$${\mathrm{\Phi }}_{\lambda }\left({sinh}^{2}r\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right).$$

For every $0<q<1$, the hypergeometric series defining ${\mathrm{\Phi }}_{\lambda }\left(t\right)$ converges normally on

$$\left\{\right(\lambda ,t):|\lambda |\le R,|t|\le q\}.$$

The same is true after any finite number of derivatives with respect to $\lambda$. Hence

$$t⟼{\mathrm{\Phi }}_{\lambda }\left(t\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)$$

is a holomorphic map from $\left|t\right|<1$ into $C_c^{\infty }\left(\mathbb{R}\right)$ with support in $[-R,R]$. Applying the inverse Opdam–Cherednik transform gives a ${\mathcal{B}}_R^{\alpha ,\beta }$-valued holomorphic map

$$\tilde{U}\left(t\right)={\mathcal{F}}_{\alpha ,\beta }^{-1}\left[{\mathrm{\Phi }}_{\lambda }\left(t\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\right].$$

For $t={sinh}^{2}r$ this coincides with ${\mathcal{M}}_r^{\alpha ,\beta }f$ by the multiplier identity. To verify the second support condition also for complex t, define

$$V\left(t\right)=R\tilde{U}\left(t\right),W\left(t\right)={\mathcal{F}}_{\alpha ,\beta }^{-1}\left[{\mathrm{\Phi }}_{\lambda }\left(t\right){\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)\left(\lambda \right)\right].$$

Both maps are holomorphic from $\left|t\right|<1$ into ${\mathcal{S}}_{\alpha ,\beta }\left(\mathbb{R}\right)$: reflection is continuous on that space and the argument used above applies to $Rf$. For real $t={sinh}^{2}r\ge 0$ sufficiently small, reflection equivariance (63) gives $V\left(t\right)=W\left(t\right)$. The identity theorem for Fréchet-valued holomorphic maps then gives $V\left(t\right)=W\left(t\right)$ throughout $\left|t\right|<1$. Therefore,

$${\mathcal{F}}_{\alpha ,\beta }\left(R\tilde{U}\left(t\right)\right)\left(\lambda \right)={\mathrm{\Phi }}_{\lambda }\left(t\right){\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)\left(\lambda \right),$$

whose support is contained in $[-R,R]$ for every complex t in the disk. Thus, both support conditions defining ${\mathcal{B}}_R^{\alpha ,\beta }$ are preserved by the holomorphic extension. Hence $\tilde{U}\left(t\right)\in {\mathcal{B}}_R^{\alpha ,\beta }$. Since ${\mathrm{\Phi }}_{\lambda }\left(0\right)=1$, we have $\tilde{U}\left(0\right)=f$. Standard holomorphic Fréchet-space calculus now gives the Taylor expansion (72), locally uniform convergence, and termwise differentiation. Finally, on the transform side, ${\mathcal{L}}_{\alpha ,\beta }$ is multiplication by $-({\lambda }^2+{\rho }^2)$; it is, therefore, a continuous endomorphism of ${\mathcal{B}}_R^{\alpha ,\beta }$ and can also be applied term by term. In particular, holomorphy here is holomorphy in the locally convex Fréchet topology; every continuous seminorm of $C_c^{\infty }\left([-R,R]\right)$ is locally bounded after multiplication by ${\mathrm{\Phi }}_{\lambda }\left(t\right)$, uniformly for t in compact subdisks, and the continuous inverse Opdam–Cherednik transform carries this property to the defining seminorms of ${\mathcal{B}}_R^{\alpha ,\beta }$. □

We may, therefore, derive the coefficient recurrence without any formal power-series assumption. With

$$t={sinh}^{2}r,$$

a direct change of variables gives

$$\begin{array}{cc}\hfill & \left[{\partial }_r^2+\left((2\alpha +1)cothr+(2\beta +1)tanhr\right){\partial }_r\right]\hfill \\ \hfill & =4t(1+t){\partial }_t^2+4\left[\alpha +1+(\rho +1)t\right]{\partial }_t.\hfill \end{array}$$

(74)

Substituting the convergent Taylor series (72) into the Darboux Equation (65), differentiating term by term, and comparing the coefficient of $t^m$ gives

$$4(m+1)(m+\alpha +1)a_{m+1}=\left[{\mathcal{L}}_{\alpha ,\beta }-4m(m+\rho )I\right]a_m.$$

(75)

Since $\alpha >-{\displaystyle \frac{1}{2}}$, all denominators $(m+1)(m+\alpha +1)$ are nonzero. Thus, the recurrence determines the Taylor coefficients uniquely from $a_0=f$. In particular, the analytic solution of the Darboux Cauchy problem is unique within ${\mathcal{B}}_R^{\alpha ,\beta }$ near $t=0$. This motivates the polynomials

$${\mathcal{P}}_0^{\alpha ,\beta }\left(z\right)=1,{\mathcal{P}}_m^{\alpha ,\beta }\left(z\right)={\displaystyle \prod _{j=0}^{m-1}}\left[z-4j(j+\rho )\right],m\ge 1.$$

(76)

Iteration of (75) yields

$$a_m={\displaystyle \frac{{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f}{4^{m}m!{(\alpha +1)}_m}}.$$

(77)

The spectral meaning of these polynomials is equally transparent:

$$\begin{array}{cc}\hfill {\mathcal{P}}_m^{\alpha ,\beta }\left(-({\lambda }^2+{\rho }^2)\right)& ={(-1)}^m{\displaystyle \prod _{j=0}^{m-1}}\left[{\lambda }^2+{(\rho +2j)}^2\right]\hfill \\ \hfill & ={(-1)}^m{4}^m{\left({\displaystyle \frac{\rho +i\lambda }{2}}\right)}_m{\left({\displaystyle \frac{\rho -i\lambda }{2}}\right)}_m.\hfill \end{array}$$

(78)

We can now state the Berenstein–Zalcman Pizzetti formula in the rank-one Opdam–Cherednik setting.

Theorem 2.

Let $f\in {\mathcal{B}}_R^{\alpha ,\beta }$. If

$$|sinhr|<1,$$

(79)

then

$${\mathcal{M}}_r^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{1}{m!{(\alpha +1)}_m}}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f.$$

(80)

Equivalently,

$${\mathcal{M}}_r^{\alpha ,\beta }f=\mathrm{\Gamma }(\alpha +1){\displaystyle \sum _{m=0}^{\infty }}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}{\displaystyle \frac{{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f}{m!\mathrm{\Gamma }(\alpha +m+1)}}.$$

(81)

The series converges in the band-limited Schwartz topology, uniformly for r in compact subsets of (79).

Proof. 

The formal expansion follows from (77). To prove convergence, apply ${\mathcal{F}}_{\alpha ,\beta }$. By (40) and (78), the transformed series is

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(-1)}^m}{m!{(\alpha +1)}_m}}{\left({sinh}^{2}r\right)}^m{\left({\displaystyle \frac{\rho +i\lambda }{2}}\right)}_m{\left({\displaystyle \frac{\rho -i\lambda }{2}}\right)}_m{\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right)\hfill \\ \hfill & ={}_{2}F_1\left({\displaystyle \frac{\rho +i\lambda }{2}},{\displaystyle \frac{\rho -i\lambda }{2}};\alpha +1;-{sinh}^{2}r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right).\hfill \end{array}$$

By (20), this equals

$${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right),$$

which is the transform of the mean by (61). The hypergeometric series converges absolutely for $|{sinh}^{2}r|<1$, and compact spectral support permits term-by-term inversion and differentiation. □

The first terms show the curvature corrections explicitly:

$$\begin{array}{cc}\hfill {\mathcal{M}}_r^{\alpha ,\beta }f=& f+{\displaystyle \frac{{sinh}^{2}r}{4(\alpha +1)}}{\mathcal{L}}_{\alpha ,\beta }f\hfill \\ \hfill & +{\displaystyle \frac{{sinh}^{4}r}{32(\alpha +1)(\alpha +2)}}{\mathcal{L}}_{\alpha ,\beta }\left[{\mathcal{L}}_{\alpha ,\beta }-4(\rho +1)I\right]f\hfill \\ \hfill & +{\displaystyle \frac{{sinh}^{6}r}{384(\alpha +1)(\alpha +2)(\alpha +3)}}{\mathcal{L}}_{\alpha ,\beta }\left[{\mathcal{L}}_{\alpha ,\beta }-4(\rho +1)I\right]\left[{\mathcal{L}}_{\alpha ,\beta }-8(\rho +2)I\right]f+O\left({sinh}^{8}r\right).\hfill \end{array}$$

(82)

In particular,

$${\left.{\partial }_r^2{\mathcal{M}}_r^{\alpha ,\beta }f\right|}_{r=0}={\displaystyle \frac{1}{2(\alpha +1)}}{\mathcal{L}}_{\alpha ,\beta }f.$$

(83)

The curvature shifts can be checked explicitly on a generalized Opdam hypergeometric eigenfunction. Since

$${\mathcal{L}}_{\alpha ,\beta }{G}_{\lambda }^{(\alpha ,\beta )}=-({\lambda }^2+{\rho }^2)G_{\lambda }^{(\alpha ,\beta )}$$

and the product formula gives

$${\mathcal{M}}_r^{\alpha ,\beta }{G}_{\lambda }^{(\alpha ,\beta )}\left(x\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)G_{\lambda }^{(\alpha ,\beta )}\left(x\right),$$

Formula (80) becomes the completely explicit scalar series

$$\begin{array}{cc}\hfill {\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)=& 1-{\displaystyle \frac{{\lambda }^2+{\rho }^2}{4(\alpha +1)}}{sinh}^{2}r\hfill \\ & +{\displaystyle \frac{({\lambda }^2+{\rho }^2)[{\lambda }^2+{(\rho +2)}^2]}{32(\alpha +1)(\alpha +2)}}{sinh}^{4}r\hfill \\ & -{\displaystyle \frac{({\lambda }^2+{\rho }^2)[{\lambda }^2+{(\rho +2)}^2][{\lambda }^2+{(\rho +4)}^2]}{384(\alpha +1)(\alpha +2)(\alpha +3)}}{sinh}^{6}r+O\left({sinh}^{8}r\right).\hfill \end{array}$$

This is exactly the beginning of the hypergeometric expansion of the Jacobi multiplier. Although a single generalized eigenfunction is not itself a vector of the band-limited Schwartz space, the example makes the shifted curvature factors transparent; Theorem 2 justifies their superposition for every $f\in {\mathcal{B}}_R^{\alpha ,\beta }$.

5. Curvature Scaling and the Rational Dunkl Limit

To display the curvature dependence explicitly, let $\kappa >0$ and replace (12) by

$$\begin{array}{cc}\hfill T_{\alpha ,\beta }^{\left(\kappa \right)}f\left(x\right)=& f^{\prime }\left(x\right)+{\displaystyle \frac{\kappa }{2}}\left((2\alpha +1)coth\left(\kappa x\right)+(2\beta +1)tanh\left(\kappa x\right)\right)\hfill \\ \hfill & \times \left(f\left(x\right)-f(-x)\right)-\kappa \rho f(-x).\hfill \end{array}$$

(84)

Set

$${\mathcal{L}}_{\alpha ,\beta }^{\left(\kappa \right)}={\left(T_{\alpha ,\beta }^{\left(\kappa \right)}\right)}^2-{\kappa }^2{\rho }^{2}I$$

(85)

and

$${\mathcal{P}}_{m,\kappa }^{\alpha ,\beta }\left(z\right)={\displaystyle \prod _{j=0}^{m-1}}\left[z-4j(j+\rho ){\kappa }^2\right].$$

(86)

The scaled mean may be written directly in terms of the unit-scale Opdam–Cherednik translation. If

$$f_{\kappa }\left(u\right)=f(u/\kappa ),$$

then

$${\tau }_{r,\kappa }^{\alpha ,\beta }f\left(x\right)=\left({\tau }_{\kappa r}^{\alpha ,\beta }{f}_{\kappa }\right)\left(\kappa x\right),$$

(87)

and hence

$${\mathcal{M}}_{r,\kappa }^{\alpha ,\beta }f\left(x\right)={\displaystyle \frac{1}{2}}\left[\left({\tau }_{\kappa r}^{\alpha ,\beta }{f}_{\kappa }\right)\left(\kappa x\right)+\left({\tau }_{-\kappa r}^{\alpha ,\beta }{f}_{\kappa }\right)\left(\kappa x\right)\right].$$

(88)

Indeed, the scaling relation

$$T_{\alpha ,\beta }^{\left(\kappa \right)}f\left(x\right)=\kappa \left[T_{\alpha ,\beta }{f}_{\kappa }\right]\left(\kappa x\right)$$

shows that (87) is the translation naturally associated with the scaled Opdam–Cherednik operator.

If $f_{\kappa }\in {\mathcal{B}}_R^{\alpha ,\beta }$, then scaling Theorem 2 gives

$${\mathcal{M}}_{r,\kappa }^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{1}{m!{(\alpha +1)}_m}}{\left({\displaystyle \frac{sinh\left(\kappa r\right)}{2\kappa }}\right)}^{2m}{\mathcal{P}}_{m,\kappa }^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }^{\left(\kappa \right)}\right)f,$$

(89)

whenever $|sinh(\kappa r\left)\right|<1$, with convergence in the correspondingly scaled band-limited Schwartz topology.

As $\kappa \downarrow 0$,

$$\kappa coth\left(\kappa x\right)⟶{\displaystyle \frac{1}{x}},\kappa tanh\left(\kappa x\right)⟶0,\kappa \rho ⟶0.$$

Therefore,

$$T_{\alpha ,\beta }^{\left(\kappa \right)}⟶D_{\alpha },D_{\alpha }f\left(x\right)=f^{\prime }\left(x\right)+{\displaystyle \frac{2\alpha +1}{2x}}\left(f\left(x\right)-f(-x)\right),$$

(90)

the rank-one rational Dunkl operator. Its square is

$$D_{\alpha }^{2}f\left(x\right)=f^{″}\left(x\right)+{\displaystyle \frac{2\alpha +1}{x}}{f}^{\prime }\left(x\right)-{\displaystyle \frac{2\alpha +1}{2x^2}}\left(f\left(x\right)-f(-x)\right).$$

(91)

The limiting mean has a concrete interpretation in terms of the rank-one Dunkl translation and spherical mean of Rösler [10]. For the rational Dunkl operator $D_{\alpha }$, the corresponding multiplicity is

$$k=\alpha +{\displaystyle \frac{1}{2}}.$$

Let ${\tau }_y^{\alpha }$ denote the one-dimensional rational Dunkl translation, characterized on Schwartz functions by

$${\mathcal{F}}_{\alpha }\left({\tau }_y^{\alpha }f\right)\left(\xi \right)=E_{\alpha +{\displaystyle \frac{1}{2}}}(iy,\xi ){\mathcal{F}}_{\alpha }f\left(\xi \right),$$

(92)

where $E_{\alpha +{\displaystyle \frac{1}{2}}}$ is the rank-one Dunkl kernel and ${\mathcal{F}}_{\alpha }$ is the rational Dunkl transform. Rösler’s one-dimensional spherical mean is

$${\mathcal{M}}_r^{\alpha }f\left(x\right)={\displaystyle \frac{1}{2}}\left({\tau }_r^{\alpha }f\left(x\right)+{\tau }_{-r}^{\alpha }f\left(x\right)\right).$$

(93)

Because the even part of the rank-one Dunkl kernel is the normalized Bessel function $j_{\alpha }$, one has

$${\mathcal{F}}_{\alpha }\left({\mathcal{M}}_r^{\alpha }f\right)\left(\xi \right)=j_{\alpha }\left(r\xi \right){\mathcal{F}}_{\alpha }f\left(\xi \right).$$

(94)

Rösler’s positivity theorem [10] gives a probability measure representation for this spherical mean. In particular, ${\mathcal{M}}_r^{\alpha }$ is positivity preserving and its representing measure is supported in the rank-one annular region

$$\left\{z\in \mathbb{R}:\left|\left|x\right|-r\right|\le \left|z\right|\le \left|x\right|+r\right\}.$$

For even functions this is the classical Bessel–Kingman spherical mean. These facts are useful for interpreting the flat limit, but the proof below needs only the multiplier identity (94).

Anker–Ayadi–Sifi prove that the rational limit of the Opdam–Cherednik product formula is precisely the rank-one Dunkl product formula [7]. After symmetrization, this identifies the product-formula limit of (88) with the Rösler mean:

$${\mathcal{M}}_{r,\kappa }^{\alpha ,\beta }f\left(x\right)⟶{\mathcal{M}}_r^{\alpha }f\left(x\right)={\displaystyle \frac{1}{2}}\left({\tau }_r^{\alpha }f\left(x\right)+{\tau }_{-r}^{\alpha }f\left(x\right)\right)(\kappa \downarrow 0),$$

(95)

for compactly supported continuous test functions and fixed $x,r$. Thus, three logically distinct contraction statements are used in this section; (95) is weak convergence tested on $C_c\left(\mathbb{R}\right)$; (97) below is fixed-order $C_{\mathrm{loc}}^{\infty }$ convergence on $C_c^{\infty }\left(\mathbb{R}\right)$; and the infinite rational series (99) is an independent identity on the rational band ${\mathcal{B}}_R^{\alpha }$. Accordingly, no single topology is asserted for all three limits; the product formula means converge weakly on compactly supported continuous test functions, each fixed shifted polynomial converges in $C_{\mathrm{loc}}^{\infty }$, and the limiting infinite series converges in the rational Dunkl Schwartz topology on a fixed compact rational spectral band. In particular, the infinite series is not obtained by interchanging $\kappa \downarrow 0$ with the curved Pizzetti sum. Thus, the topology is stated explicitly at every use; weak convergence is understood against $C_c\left(\mathbb{R}\right)$ test functions in (95); fixed-order operator convergence is in $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$ in (97); the rational Pizzetti series converges in the rational Dunkl Schwartz topology; and the subsequent $L_{\alpha }^2$ convergence is only a continuous consequence of Schwartz convergence, not an additional curvature-contraction topology.

For $R>0$, let

$${\mathcal{B}}_R^{\alpha }=\left\{f\in \mathcal{S}\left(\mathbb{R}\right):supp{\mathcal{F}}_{\alpha }f\subset [-R,R]\right\}.$$

(96)

The contraction statement below is deliberately separated into two parts. First, for each fixed polynomial order we prove convergence in the concrete topology $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$. Second, the infinite rational Pizzetti series is proved independently from the rational Dunkl multiplier. We, therefore, do not pass to the limit $\kappa \downarrow 0$ term by term through the infinite series (89), and no uniform-in-$\kappa$ summability assertion is required.

Theorem 3.

For every fixed m and every $f\in C_c^{\infty }\left(\mathbb{R}\right)$,

$${\mathcal{P}}_{m,\kappa }^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }^{\left(\kappa \right)}\right)f⟶{\left(D_{\alpha }^2\right)}^{m}f$$

(97)

in $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$ as $\kappa \downarrow 0$. Moreover,

$${\displaystyle \frac{sinh\left(\kappa r\right)}{2\kappa }}⟶{\displaystyle \frac{r}{2}}.$$

(98)

Thus, the coefficients in (89) contract to the rational Dunkl coefficients. If $f\in {\mathcal{B}}_R^{\alpha }$, then for every $r\ge 0$,

$${\mathcal{M}}_r^{\alpha }f={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(r/2)}^{2m}}{m!{(\alpha +1)}_m}}{\left(D_{\alpha }^2\right)}^{m}f,$$

(99)

with convergence in the Schwartz topology and hence in the rational Dunkl $L^2$ space. Equivalently, (99) is the Taylor expansion of the Bessel multiplier $j_{\alpha }\left(r\xi \right)$ in (94).

Proof. 

For $f\in C_c^{\infty }\left(\mathbb{R}\right)$, the quotient

$${\displaystyle \frac{f\left(x\right)-f(-x)}{x}}$$

extends smoothly across the origin. The Taylor expansions of $\kappa coth\left(\kappa x\right)$ and $\kappa tanh\left(\kappa x\right)$, therefore, imply

$$T_{\alpha ,\beta }^{\left(\kappa \right)}f⟶D_{\alpha }f$$

in $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$. Iteration gives ${\mathcal{L}}_{\alpha ,\beta }^{\left(\kappa \right)}f\to D_{\alpha }^{2}f$, and for each fixed j,

$$\left[{\mathcal{L}}_{\alpha ,\beta }^{\left(\kappa \right)}-4j(j+\rho ){\kappa }^2\right]f⟶D_{\alpha }^{2}f.$$

The coefficient functions of these differential–reflection operators converge with all derivatives on compact sets after the removable singularities are filled in. Finite compositions, therefore, converge on $C_c^{\infty }\left(\mathbb{R}\right)$ by induction. Since m is fixed, this proves (97); (98) is immediate.

For the infinite limiting series, let $f\in {\mathcal{B}}_R^{\alpha }$. On the rational Dunkl-transform side,

$${\mathcal{F}}_{\alpha }\left(D_{\alpha }^{2}f\right)\left(\xi \right)=-{\xi }^2{\mathcal{F}}_{\alpha }f\left(\xi \right),$$

while

$$j_{\alpha }\left(r\xi \right)=\mathrm{\Gamma }(\alpha +1){\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(-1)}^m{(r\xi /2)}^{2m}}{m!\mathrm{\Gamma }(\alpha +m+1)}}.$$

This Bessel series and all its $\xi$-derivatives converge uniformly on the compact band $\left|\xi \right|\le R$. Hence termwise inversion is valid in the Schwartz topology, and the multiplier of the right-hand side of (99) is exactly $j_{\alpha }\left(r\xi \right)$. The identity now follows from (94). We also spell out the implication from Schwartz convergence to the rational Dunkl $L^2$ convergence used in the statement. Up to the harmless normalization constant, the latter norm is

$${\parallel h\parallel }_{L_{\alpha }^2}^2={\int }_{\mathbb{R}}{\left|h\left(x\right)\right|}^2{\left|x\right|}^{2\alpha +1}\mathrm{d}x.$$

Choose an integer $N>\alpha +1$ and the Schwartz seminorm $p_N\left(h\right)={sup}_x{(1+|x\left|\right)}^N\left|h\left(x\right)\right|$. Then

$${\parallel h\parallel }_{L_{\alpha }^2}^2\le p_N{\left(h\right)}^2{\int }_{\mathbb{R}}{(1+|x\left|\right)}^{-2N}{\left|x\right|}^{2\alpha +1}\mathrm{d}x,$$

and the last integral is finite because $\alpha >-{\displaystyle \frac{1}{2}}$ and $N>\alpha +1$. Hence convergence in the rational Dunkl Schwartz topology implies convergence in $L_{\alpha }^2$ continuously. □

The parameter $\beta$ disappears in the flat limit. This reflects the fact that the tanh part is regular at the origin, whereas the coth singularity is the part that survives the contraction. This also clarifies the logical status of (99); it is not obtained by an unproved uniform limit of the curved infinite expansions. Only each fixed shifted polynomial contracts in $C_{\mathrm{loc}}^{\infty }$, while the full limiting series follows from the uniformly convergent Bessel multiplier series on compact rational Dunkl spectral bands.

Remark 1

(Why $\kappa \to \infty$ is not the compact dual). The scaling used above varies the magnitude of the negative curvature inside the noncompact hyperbolic/Jacobi model. At fixed $x\ne 0$, letting $\kappa \to \infty$ in the coefficients $\kappa coth\left(\kappa x\right)$ and $\kappa tanh\left(\kappa x\right)$ does not produce the radial operator of a compact rank-one symmetric space. The compact dual is obtained instead by the analytic continuation of the radial coordinate (hyperbolic functions are replaced by trigonometric ones), leading to a finite radial interval, Jacobi polynomials, and a discrete spherical spectrum. Thus, it is not a third limit of (84) within the present real-line Opdam–Cherednik transform. A compact-type Pizzetti analogue should nevertheless be possible in the compact Heckman–Opdam picture, with the same Darboux-coefficient but a different translation structure and spectral measure. Developing that compact dual theory is beyond the scope of the present paper.

6. Rank-One Symmetric Spaces and Two-Radius Stability

We now identify the real hyperbolic specialization of the Opdam–Cherednik Pizzetti formula and its relation with the classical Berenstein–Zalcman expansion. Let ${\mathbb{H}}^n$, $n\ge 2$, denote the real hyperbolic space of constant sectional curvature $-1$. Fix $o\in {\mathbb{H}}^n$ and write geodesic polar coordinates around o as

$$x={exp}_o\left(s\omega \right),s=d(o,x),\omega \in {\mathbb{S}}^{n-1}.$$

The hyperbolic metric and Riemannian volume element are

$$d{s}_{{\mathbb{H}}^n}^2=d{s}^2+{sinh}^{2}sd{\omega }^2,d{V}_{{\mathbb{H}}^n}={(sinhs)}^{n-1}\mathrm{d}s\mathrm{d}\omega .$$

(100)

Consequently,

$${\mathrm{\Delta }}_{{\mathbb{H}}^n}={\displaystyle \frac{{\partial }^2}{\partial s^2}}+(n-1)coths{\displaystyle \frac{\partial }{\partial s}}+{\displaystyle \frac{1}{{sinh}^{2}s}}{\mathrm{\Delta }}_{{\mathbb{S}}^{n-1}},$$

(101)

where ${\mathrm{\Delta }}_{{\mathbb{S}}^{n-1}}$ is the Laplace–Beltrami operator on the unit sphere; see [11,12].

If f is radial about o, say

$$f\left(x\right)=u\left(d(o,x)\right)=u\left(s\right),$$

then the angular term in (101) vanishes and

$${\mathrm{\Delta }}_{{\mathbb{H}}^n}f\left(x\right)=u^{″}\left(s\right)+(n-1)coths{u}^{\prime }\left(s\right).$$

(102)

It is, therefore, natural to introduce the radial Laplace–Beltrami operator

$$L_n={\displaystyle \frac{d^2}{d{s}^2}}+(n-1)coths{\displaystyle \frac{d}{ds}}.$$

(103)

For $f\in C^{\infty }\left({\mathbb{H}}^n\right)$, the normalized geodesic spherical mean of radius r centered at x is

$$M_r^{{\mathbb{H}}^n}f\left(x\right)={\displaystyle \frac{1}{|{\mathbb{S}}^{n-1}|}}{\int }_{{\mathbb{S}}^{n-1}}f\left({exp}_x\left(r\eta \right)\right)\mathrm{d}\eta ,r\ge 0.$$

(104)

It satisfies

$$M_0^{{\mathbb{H}}^n}f=f.$$

For radial functions, the spherical mean reduces to an explicit one-dimensional integral. Let

$$f\left(y\right)=u\left(d(o,y)\right),s=d(o,x).$$

If $\theta$ denotes the angle at x between the geodesic segment joining x to o and the geodesic segment joining x to ${exp}_x\left(r\eta \right)$, then the hyperbolic law of cosines gives

$$coshd\left(o,{exp}_x\left(r\eta \right)\right)=coshscoshr-sinhssinhrcos\theta .$$

(105)

Hence

$$\begin{array}{cc}\hfill M_r^{{\mathbb{H}}^n}f\left(x\right)=& {\displaystyle \frac{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{n-1}{2}\right)}}\hfill \\ \hfill & \times {\int }_0^{\pi }u\left(arcosh\left(coshscoshr-sinhssinhrcos\theta \right)\right){(sin\theta )}^{n-2}\mathrm{d}\theta .\hfill \end{array}$$

(106)

Equivalently, after setting $t=cos\theta$,

$$\begin{array}{cc}\hfill M_r^{{\mathbb{H}}^n}f\left(x\right)=& {\displaystyle \frac{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{n-1}{2}\right)}}\hfill \\ \hfill & \times {\int }_{-1}^{1}u\left(arcosh\left(coshscoshr-sinhssinhrt\right)\right){(1-t^2)}^{{\displaystyle \frac{n-3}{2}}}\mathrm{d}t.\hfill \end{array}$$

(107)

The normalization follows from

$${\displaystyle \frac{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{n-1}{2}\right)}}{\int }_{-1}^1{(1-t^2)}^{{\displaystyle \frac{n-3}{2}}}\mathrm{d}t=1.$$

In particular, at the center $x=o$ one has $s=0$, and, therefore,

$$M_r^{{\mathbb{H}}^n}f\left(o\right)=u\left(r\right).$$

(108)

The spherical-mean family satisfies the Darboux transmutation identity

$$\begin{array}{cc}\hfill & \left[{\partial }_r^2+(n-1)cothr{\partial }_r\right]M_r^{{\mathbb{H}}^n}f\left(x\right)\hfill \\ \hfill & ={\mathrm{\Delta }}_{{\mathbb{H}}^n,x}{M}_r^{{\mathbb{H}}^n}f\left(x\right)=M_r^{{\mathbb{H}}^n}\left({\mathrm{\Delta }}_{{\mathbb{H}}^n}f\right)\left(x\right),r>0,\hfill \end{array}$$

(109)

with

$$M_0^{{\mathbb{H}}^n}f\left(x\right)=f\left(x\right),{\left.{\partial }_r{M}_r^{{\mathbb{H}}^n}f\left(x\right)\right|}_{r=0}=0.$$

(110)

Thus, the differential operator acting on the mean radius r in (109) has precisely the same form as the radial Laplace–Beltrami operator $L_n$.

We now compare this structure with the rank-one Opdam–Cherednik setting. Take

$$\alpha ={\displaystyle \frac{n}{2}}-1,\beta =-{\displaystyle \frac{1}{2}}.$$

(111)

Then

$$\rho =\alpha +\beta +1={\displaystyle \frac{n-1}{2}}.$$

(112)

If the radial profile u is regarded as an even function on $\mathbb{R}$, then the reflection term disappears and the even restriction of the Opdam–Cherednik Laplacian becomes

$${\left.{\mathcal{L}}_{\alpha ,\beta }\right|}_{\mathrm{even}}={\displaystyle \frac{d^2}{d{s}^2}}+(n-1)coths{\displaystyle \frac{d}{ds}}=L_n.$$

(113)

Moreover,

$$4j(j+\rho )=4j\left(j+{\displaystyle \frac{n-1}{2}}\right)=2j(2j+n-1).$$

(114)

Thus, both the differential operator and the shifted factors in the Opdam–Cherednik Pizzetti polynomials reduce exactly to their real-hyperbolic radial counterparts.

For radial functions, the symmetrized Opdam–Cherednik mean with parameters (111) agrees with the radial hyperbolic spherical mean (106). In order to invoke Theorem 2 without enlarging its hypotheses, we state the geometric corollary on the corresponding compact spectral band.

Corollary 2.

Let

$$f\left(x\right)=u\left(d\right(o,x\left)\right)$$

be a smooth radial function on ${\mathbb{H}}^n$, and suppose that the even extension of u belongs to ${\mathcal{B}}_R^{{\displaystyle \frac{n}{2}}-1,-{\displaystyle \frac{1}{2}}}$ for some $R>0$. At the center we also record the standard smooth-radial compatibility condition

$$u^{(2k+1)}\left(0\right)=0,k=0,1,2,\dots ,$$

which is automatic for a smooth even extension. Then, for $s=d(o,x)$ and every r satisfying $|sinhr|<1$,

$$\begin{array}{cc}\hfill M_r^{{\mathbb{H}}^n}f\left(x\right)=& u\left(s\right)+\mathrm{\Gamma }\left({\displaystyle \frac{n}{2}}\right){\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \frac{1}{m!\mathrm{\Gamma }(\frac{n}{2}+m)}}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}\hfill \\ \hfill & \times L_n{\displaystyle \prod _{j=1}^{m-1}}\left[L_n-2j(2j+n-1)I\right]u\left(s\right).\hfill \end{array}$$

(115)

The series converges in the band-limited Schwartz topology of the even profile and hence in $C^{\infty }$ on every compact set of the radial variable s. The radius restriction is the explicit condition $|sinhr|<1$ inherited from Theorem 2; it is independent of the point s and of the particular profile inside the fixed spectral band. For $m=1$, the empty product is understood as the identity.

Indeed, under (111),

$$\alpha +1={\displaystyle \frac{n}{2}},\rho ={\displaystyle \frac{n-1}{2}},{\left.{\mathcal{L}}_{\alpha ,\beta }\right|}_{\mathrm{even}}=L_n,$$

and, therefore,

$${(\alpha +1)}_m={\left({\displaystyle \frac{n}{2}}\right)}_m={\displaystyle \frac{\mathrm{\Gamma }(\frac{n}{2}+m)}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}}$$

and

$${\mathcal{P}}_m^{\alpha ,\beta }\left(L_n\right)={\displaystyle \prod _{j=0}^{m-1}}\left[L_n-2j(2j+n-1)I\right].$$

Consequently, the general Opdam–Cherednik Pizzetti expansion

$${\mathcal{M}}_r^{\alpha ,\beta }u={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(sinhr/2)}^{2m}}{m!{(\alpha +1)}_m}}{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)u$$

reduces precisely to (115). In this sense, the Berenstein–Zalcman shifted Laplace–Beltrami products are the real-hyperbolic radial specialization of the rank-one Opdam–Cherednik Pizzetti polynomials.

The preceding correspondence is structural rather than merely coefficient-wise. Hyperbolic spherical averaging intertwines the Jacobi radial operator in the mean radius with the Laplace–Beltrami operator, whereas the symmetrized Opdam–Cherednik mean intertwines the same Jacobi-type radial operator with ${\mathcal{L}}_{\alpha ,\beta }$. Under (111), these two mechanisms coincide on the radial subspace.

The real-hyperbolic case is one member of the full rank-one noncompact symmetric-space family. We now record this specialization in root-multiplicity form. Let

$$X=G/K$$

be a Riemannian symmetric space of noncompact type and rank one, and let $m_{\gamma }$ and $m_{2\gamma }$ denote the multiplicities of the positive restricted roots $\gamma$ and $2\gamma$, respectively. With the standard normalization of the geodesic distance, the radial part of the Laplace–Beltrami operator is

$${\mathrm{\Delta }}_X^{\mathrm{rad}}={\displaystyle \frac{d^2}{d{s}^2}}+\left[(m_{\gamma }+m_{2\gamma })coths+m_{2\gamma }tanhs\right]{\displaystyle \frac{d}{ds}}.$$

(116)

Indeed, using

$$2coth\left(2s\right)=coths+tanhs,$$

this is the usual radial expression associated with the restricted root system of type $B{C}_1$.

Comparing (116) with the even part of the Opdam–Cherednik Laplacian,

$${\left.{\mathcal{L}}_{\alpha ,\beta }\right|}_{\mathrm{even}}={\displaystyle \frac{d^2}{d{s}^2}}+\left[(2\alpha +1)coths+(2\beta +1)tanhs\right]{\displaystyle \frac{d}{ds}},$$

gives the parameter correspondence

$$\alpha ={\displaystyle \frac{m_{\gamma }+m_{2\gamma }-1}{2}},\beta ={\displaystyle \frac{m_{2\gamma }-1}{2}}.$$

(117)

Consequently,

$$\rho =\alpha +\beta +1={\displaystyle \frac{m_{\gamma }+2m_{2\gamma }}{2}},$$

(118)

which is the usual half-sum of the positive restricted roots counted with multiplicities.

Thus, the radial harmonic analysis on every rank-one noncompact symmetric space is obtained from the even Opdam–Cherednik theory by the specialization (117). In particular, the four geometric families correspond to

$\mathit{X}$	${\mathit{m}}_{\mathit{\gamma }}$	${\mathit{m}}_{\mathbf{2}\mathit{\gamma }}$	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\rho }$
${\mathbb{H}}_{\mathbb{R}}^n$	$n-1$	0	${\displaystyle \frac{n}{2}}-1$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{n-1}{2}}$
${\mathbb{H}}_{\mathbb{C}}^n$	$2(n-1)$	1	$n-1$	0	n
${\mathbb{H}}_{\mathbb{H}}^n$	$4(n-1)$	3	$2n-1$	1	$2n+1$
${\mathbb{H}}_{\mathbb{O}}^2$	8	7	7	3	11

Here ${\mathbb{H}}_{\mathbb{O}}^2$ denotes the Cayley hyperbolic plane. The table has been checked against (117)–(118); in each row

$$\alpha ={\displaystyle \frac{m_{\gamma }+m_{2\gamma }-1}{2}},\beta ={\displaystyle \frac{m_{2\gamma }-1}{2}},\rho =\alpha +\beta +1$$

with the same normalization of the geodesic variable used in (116). These parameter identifications are the standard ones in the Jacobi-function realization of spherical analysis on rank-one symmetric spaces; see [5,6,11].

For a radial function

$$f\left(x\right)=u\left(d\right(o,x\left)\right)$$

on X, write

$$L_X={\mathrm{\Delta }}_X^{\mathrm{rad}}.$$

For the Pizzetti expansion below we assume that the even extension of u belongs to ${\mathcal{B}}_R^{\alpha ,\beta }$ for some $R>0$, with $\alpha ,\beta$ given by (117). At the center we impose the usual smooth-radial compatibility $u^{(2k+1)}\left(0\right)=0$, which is automatic for the smooth even extension. This compact spectral support hypothesis is exactly the one used in Theorem 2; no assertion is made here for an arbitrary real-analytic radial profile. Thus, the displayed symmetric-space formula is stated only under the same compact spectral-band hypothesis as Theorem 2. A broader local statement is available when the even radial profile is known independently to be locally Pizzetti-admissible in the sense of Definition 1; mere real analyticity by itself is not used as a sufficient hypothesis in this paper. The normalized geodesic spherical mean is then the Jacobi spherical mean corresponding to the parameters (117). Hence the Opdam–Cherednik Pizzetti formula gives directly

$$\begin{array}{cc}\hfill M_r^{X}f\left(x\right)=& u\left(s\right)+\mathrm{\Gamma }(\alpha +1){\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \frac{1}{m!\mathrm{\Gamma }(\alpha +m+1)}}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}\hfill \\ \hfill & \times L_X{\displaystyle \prod _{j=1}^{m-1}}\left[L_X-4j(j+\rho )I\right]u\left(s\right),s=d(o,x),\hfill \end{array}$$

(119)

for $|sinhr|<1$, with convergence in the band-limited Schwartz topology and, therefore, in $C^{\infty }$ on compact sets of $s=d(o,x)$. Thus, the admissible radius is explicit and does not depend on the point s or on the particular profile in the fixed spectral band. The condition $|sinhr|<1$ is the local power-series restriction in this symmetric-space specialization as well. It is inherited from Theorem 2; it does not restrict the geometric definition of the spherical mean $M_r^X$, which is defined for all admissible geodesic radii independently of the local Pizzetti expansion. Thus, the complete sequence of Berenstein–Zalcman shifts on a rank-one symmetric space is encoded by

$$4j(j+\rho )=2j\left(2j+m_{\gamma }+2m_{2\gamma }\right).$$

(120)

Formula (119) shows that the Opdam–Cherednik Pizzetti expansion is not merely an analogue of the symmetric-space formula. Its even specialization contains the radial Pizzetti calculus of every rank-one noncompact Riemannian symmetric space. The reflection term in the full Opdam–Cherednik operator, therefore, provides a nonsymmetric differential–reflection extension of the classical radial theory.

We now return to the general Opdam–Cherednik setting. A further application of the multiplier representation concerns uniqueness and stability from two generalized radii. This is related to the two-radius and Pompeiu phenomena considered by Berenstein and Zalcman [2,3]. On a compact spectral band, the absence of common zeros of the corresponding Jacobi-function multipliers yields a quantitative stability estimate.

For $R>0$ and $r_1,r_2>0$, define

$${\delta }_R(r_1,r_2)=\underset{\left|\lambda \right|\le R}{min}{\left({\left|{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_1\right)\right|}^2+{\left|{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_2\right)\right|}^2\right)}^{1/2}.$$

(121)

For real $\lambda$ and real r, the Jacobi function ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)$ is real-valued under (10); the two upper hypergeometric parameters are complex conjugates and the argument $-{sinh}^{2}r$ is real. We nevertheless keep absolute values in (121), since that formulation is invariant under harmless changes of spectral normalization.

Theorem 4.

Let

$$f\in {\mathcal{B}}_R^{\alpha ,\beta }$$

and let $r_1,r_2>0$. Assume that

$$\left\{\lambda \in [-R,R]:{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_1\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_2\right)=0\right\}=⌀.$$

(122)

Then

$${\delta }_R(r_1,r_2)>0$$

and

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{L_{\alpha ,\beta }^2}\le & {\displaystyle \frac{1}{{\delta }_R(r_1,r_2)}}\hfill \\ \hfill & \times {\left({∥{\mathcal{M}}_{r_1}^{\alpha ,\beta }f∥}_{L_{\alpha ,\beta }^2}^2+{∥{\mathcal{M}}_{r_2}^{\alpha ,\beta }f∥}_{L_{\alpha ,\beta }^2}^2\right)}^{1/2}.\hfill \end{array}$$

(123)

In particular,

$${\mathcal{M}}_{r_1}^{\alpha ,\beta }f={\mathcal{M}}_{r_2}^{\alpha ,\beta }f=0⟹f=0.$$

Proof. 

The function

$$\lambda ⟼{\left({\left|{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_1\right)\right|}^2+{\left|{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_2\right)\right|}^2\right)}^{1/2}$$

is continuous on the compact interval $[-R,R]$. Condition (122), therefore, implies

$${\delta }_R(r_1,r_2)>0.$$

By the multiplier identity (61),

$${\mathcal{F}}_{\alpha ,\beta }\left({\mathcal{M}}_{r_j}^{\alpha ,\beta }f\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_j\right){\mathcal{F}}_{\alpha ,\beta }f\left(\lambda \right),j=1,2.$$

Moreover, since ${\mathcal{M}}_{r_j}^{\alpha ,\beta }$ commutes with reflection,

$${\mathcal{F}}_{\alpha ,\beta }\left(R{\mathcal{M}}_{r_j}^{\alpha ,\beta }f\right)\left(\lambda \right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_j\right){\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)\left(\lambda \right).$$

Applying the Plancherel identity to the two means and adding gives

$$\begin{array}{cc}& {∥{\mathcal{M}}_{r_1}^{\alpha ,\beta }f∥}_{L_{\alpha ,\beta }^2}^2+{∥{\mathcal{M}}_{r_2}^{\alpha ,\beta }f∥}_{L_{\alpha ,\beta }^2}^2\hfill \\ & \ge {\delta }_R{(r_1,r_2)}^2{\parallel f\parallel }_{L_{\alpha ,\beta }^2}^2.\hfill \end{array}$$

Taking square roots proves (123). The final assertion follows immediately. □

Remark 2.

Condition (122) is not restrictive near the origin. Indeed, ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(0\right)=1$ for every real λ, and $(\lambda ,r)\mapsto {\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)$ is continuous on compact sets. Hence, for every fixed $R>0$, there exists $r_{*}\left(R\right)>0$ such that

$$\underset{\left|\lambda \right|\le R}{inf}\left|{\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)\right|\ge {\displaystyle \frac{1}{2}},0\le r\le r_{*}\left(R\right).$$

Consequently every pair

$$(r_1,r_2)=(\epsilon ,2\epsilon ),0<2\epsilon <r_{*}\left(R\right),$$

satisfies (122); in fact one sufficiently small radius already prevents a common zero, while the second radius may be chosen arbitrarily. The number $r_{*}\left(R\right)$ is readily computable from the continuous Jacobi multiplier, or bounded from the uniform local series beginning with

$${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)=1-{\displaystyle \frac{{\lambda }^2+{\rho }^2}{4(\alpha +1)}}{sinh}^{2}r+O_R\left({sinh}^{4}r\right).$$

Remark 3.

The compact spectral-band hypothesis has a precise role; it converts the absence of common zeros of ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_1\right)$ and ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r_2\right)$ into the uniform lower bound ${\delta }_R(r_1,r_2)>0$. Extending the two-radius assertion to larger function or distribution spaces is a different problem, requiring spectral synthesis and a finer analysis of the common zero sets of Jacobi functions, as in the classical Berenstein–Zalcman theory of the Pompeiu problem [2,3].

7. Applications

7.1. Mean-Value, Shifted-Polyharmonic, and Ball-Mean Principles

Pizzetti expansions are useful not only as local series formulas but also as tools for characterizing harmonicity, finite polyharmonicity, and geometric mean-value properties. In the classical Euclidean and Riemannian settings these themes go back to Pizzetti and appear prominently in mean-value theory and in the Berenstein–Zalcman approach to the Pompeiu problem; see [1,2,3,11,12]. The purpose of this section is to show that the shifted Opdam–Cherednik Pizzetti polynomials obtained in Section 4 lead to analogous applications without discarding the reflection structure. We treat three consequences: a mean-value characterization of ${\mathcal{L}}_{\alpha ,\beta }$-harmonic functions, a finite shifted-polyharmonic principle, and a normalized ball-mean expansion. To formulate the first two consequences without imposing an unnecessary $L^2$ restriction, we use the following local analytic class.

Definition 1.

A function $f\in C^{\infty }\left(\mathbb{R}\right)$ is called locally Pizzetti-admissible (relative to the symmetrized mean and the Darboux flow) if there exists $r_0>0$ such that ${\mathcal{M}}_r^{\alpha ,\beta }f$ is defined for $0\le r<r_0$, admits the natural even extension in r, and the map

$$t⟼{\mathcal{M}}_{arsinh\sqrt{t}}^{\alpha ,\beta }f,t\ge 0,$$

admits a $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$-valued analytic extension to a neighborhood of $t=0$. We also require that this extension satisfy the Darboux Equation (65) for real r near the origin. Here the Darboux equation is required classically, not merely in the distributional sense; after the removable singularity at $r=0$ is filled in, both sides of (65) are equal as $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$ functions of the spatial variable, and the r-derivatives are taken in the $C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$ Fréchet topology.

Definition 1 is a conditional admissibility hypothesis. It does not assert that every $C^{\infty }$ function admits a generalized mean with the stated analytic dependence, nor that the Darboux equation follows from smoothness alone. Accordingly, every characterization below is to be understood within the class of locally Pizzetti-admissible functions. Thus, for example, the mean-value characterization says that harmonicity is equivalent to local invariance of the symmetrized mean provided the admissibility hypotheses are already satisfied; it is not a converse criterion for admissibility itself. The same qualification applies to the finite shifted-polyharmonic characterization. No characterization of all smooth functions satisfying the admissibility hypotheses is claimed. Here “$C_{\mathrm{loc}}^{\infty }\left(\mathbb{R}\right)$-valued analytic” is meant in the standard Fréchet sense. For a compact set $K⋐\mathbb{R}$ and $q\in {\mathbb{N}}_0$, put

$$p_{K,q}\left(h\right)=\underset{0\le j\le q}{max}\underset{x\in K}{sup}\left|{\partial }_x^{j}h\left(x\right)\right|.$$

An extension $V\left(t\right)$ is analytic at 0 if $V\left(t\right)={\sum }_{m\ge 0}{v}_m{t}^m$ in a disk $\left|t\right|<\eta$ and, for every $K,q$ and every $0<{\eta }^{\prime }<\eta$,

$${\displaystyle \sum _{m=0}^{\infty }}{p}_{K,q}\left(v_m\right){\left({\eta }^{\prime }\right)}^m<\infty .$$

Thus, the Taylor series converges locally uniformly with every spatial derivative.

Besides the compact spectral-band vectors, the class contains explicit non-band-limited, nonconstant functions. For every real $\lambda$, the Opdam hypergeometric eigenfunction $G_{\lambda }^{(\alpha ,\beta )}$ obeys the product formula

$${\mathcal{M}}_r^{\alpha ,\beta }{G}_{\lambda }^{(\alpha ,\beta )}\left(x\right)={\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)G_{\lambda }^{(\alpha ,\beta )}\left(x\right).$$

Since ${\phi }_{\lambda }^{(\alpha ,\beta )}\left(r\right)={\mathrm{\Phi }}_{\lambda }\left({sinh}^{2}r\right)$ with ${\mathrm{\Phi }}_{\lambda }$ holomorphic for $|{sinh}^{2}r|<1$, this identity verifies the analytic condition and the Darboux equation directly. For real $\lambda$ these generalized eigenfunctions belong to the continuous spectral family and are not vectors of any compact-band Schwartz class ${\mathcal{B}}_R^{\alpha ,\beta }$. Taking, for instance, $\lambda =0$ gives a concrete nonconstant example (the constant spectral parameter is $\lambda =i\rho$, not a real parameter under the standing assumption $\rho >0$). Every $f\in {\mathcal{B}}_R^{\alpha ,\beta }$ is locally Pizzetti-admissible by Proposition 2. The point of Definition 1 is that it also permits natural non-$L^2$ solutions, such as constants and other local solutions of the Opdam–Cherednik differential equations. The class in Definition 1 is not proposed as a replacement for the Schwartz, Sobolev, or Paley–Wiener spaces. It is a local analytic-vector condition for the Darboux flow in the variable $t={sinh}^{2}r$. The inclusions proved here are the ones needed below; every compact spectral-band vector belongs to this class, while the local class also contains analytic solutions that need not lie in the global Opdam–Cherednik $L^2$ theory. We make no claim of an abstract characterization of this class in a standard Sobolev scale; its role is only to state local mean-value and finite shifted-polyharmonic consequences without imposing artificial global integrability. From the evolution viewpoint, Definition 1 may be read as imposing an analytic-vector condition on the symmetrized Darboux flow near $t=0$. This is an interpretation of the defining hypotheses, not an independent characterization of all smooth functions. Two readily verifiable sufficient classes used in the paper are (i) every compact spectral-band vector $f\in {\mathcal{B}}_R^{\alpha ,\beta }$ and (ii) the generalized eigenfunctions $G_{\lambda }^{(\alpha ,\beta )}$ (and their finite linear combinations), for which the product formula reduces the mean to the scalar Jacobi multiplier. These examples show that the admissible class extends beyond the global band-limited $L^2$ framework, without asserting that the listed sufficient classes exhaust it.

Corollary 3.

Within the class of locally Pizzetti-admissible functions, let f be given. Then

$${\mathcal{L}}_{\alpha ,\beta }f=0$$

(124)

if and only if

$${\mathcal{M}}_r^{\alpha ,\beta }f=f$$

(125)

for every sufficiently small r.

Proof. 

Write

$${\mathcal{M}}_{arsinh\sqrt{t}}^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{\infty }}{a}_m{t}^m.$$

The Darboux equation gives the recurrence (75), hence

$$a_m={\displaystyle \frac{{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f}{4^{m}m!{(\alpha +1)}_m}}.$$

If (124) holds, then ${\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f=0$ for every $m\ge 1$, because every such polynomial contains the factor ${\mathcal{L}}_{\alpha ,\beta }$. Thus, the mean equals f. Conversely, (125) implies ${\partial }_r^2{\mathcal{M}}_r^{\alpha ,\beta }{f|}_{r=0}=0$, and (83) gives (124). □

Within the admissible class fixed above, this is the Opdam–Cherednik counterpart of the classical principle that harmonic functions are characterized by invariance under spherical averaging. Here the averaging is generated by the symmetrized Opdam–Cherednik translation rather than by an ordinary group action, and the infinitesimal operator is the reflection–differential operator ${\mathcal{L}}_{\alpha ,\beta }$. The full shifted products give an analogous conditional characterization of finite radial complexity within the same admissible class.

Theorem 5.

Let $N\ge 1$. Within the class of locally Pizzetti-admissible functions, the following conditions on f are equivalent:

 (i) 

$${\mathcal{P}}_N^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f=0.$$

(126)

 (ii) 

The map

$$t⟼{\mathcal{M}}_{arsinh\sqrt{t}}^{\alpha ,\beta }f$$

(127)

is a polynomial in t of degree at most $N-1$ near $t=0$.

When these conditions hold,

$${\mathcal{M}}_r^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{N-1}}{\displaystyle \frac{{(sinhr/2)}^{2m}}{m!{(\alpha +1)}_m}}{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f$$

(128)

for all sufficiently small r.

Proof. 

For $m\ge N$,

$${\mathcal{P}}_m^{\alpha ,\beta }\left(z\right)={\mathcal{P}}_N^{\alpha ,\beta }\left(z\right){\displaystyle \prod _{j=N}^{m-1}}[z-4j(j+\rho )].$$

Hence (126) annihilates every coefficient of order $m\ge N$ in the local Pizzetti series and yields (128). Conversely, if (127) has degree at most $N-1$, its $t^N$ coefficient is zero. Formula (77) then gives (126). □

Remark 4.

The distinction between the local admissible class and the band-limited $L^2$ class is essential. Indeed, for real λ,

$${\mathcal{P}}_N^{\alpha ,\beta }\left(-({\lambda }^2+{\rho }^2)\right)={(-1)}^N{\displaystyle \prod _{j=0}^{N-1}}\left[{\lambda }^2+{(\rho +2j)}^2\right]\ne 0.$$

(129)

Consequently, if $f\in {\mathcal{B}}_R^{\alpha ,\beta }$ and ${\mathcal{P}}_N^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f=0$, then $f=0$ by injectivity of the transform. The nontrivial content of Theorem 5, therefore, concerns the natural local or non-$L^2$ analytic classes, exactly as mean-value principles for constant or harmonic functions do in noncompact geometry.

A concrete nontrivial local example is obtained from the analytic continuation of the Opdam hypergeometric eigenfunctions. Let $N\ge 2$ and choose $1\le j\le N-1$. With

$$f_j\left(x\right)=G_{i(\rho +2j)}^{(\alpha ,\beta )}\left(x\right),$$

Equation (26) gives

$${\mathcal{L}}_{\alpha ,\beta }{f}_j=4j(j+\rho )f_j.$$

Consequently,

$${\mathcal{P}}_N^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f_j=0,$$

because the factor with index j vanishes. By analytic continuation in the spectral parameter of the product formula, $f_j$ is locally Pizzetti-admissible. For $j\ge 1$ it is nonconstant and it lies outside the real band-limited $L^2$ spectral class, providing an example within the conditional scope of Theorem 5. A volumetric version follows by averaging the spherical means with the Jacobi radial density. Define

$${\mathcal{B}}_r^{\alpha ,\beta }f={\displaystyle \frac{{\displaystyle {\int }_0^r{\mathcal{M}}_s^{\alpha ,\beta }f{(sinhs)}^{2\alpha +1}{(coshs)}^{2\beta +1}\mathrm{d}s}}{{\displaystyle {\int }_0^r{(sinhs)}^{2\alpha +1}{(coshs)}^{2\beta +1}\mathrm{d}s}}}.$$

(130)

This normalized operator is the Jacobi analogue of averaging over a geodesic ball: the denominator is precisely the radial volume factor associated with the Jacobi weight.

Proposition 3.

Let $f\in {\mathcal{B}}_R^{\alpha ,\beta }$. For $r>0$ with $|sinhr|<1$,

$${\mathcal{B}}_r^{\alpha ,\beta }f={\displaystyle \sum _{m=0}^{\infty }}{c}_m^{\alpha ,\beta }\left(r\right){\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right)f,$$

(131)

where

$$c_m^{\alpha ,\beta }\left(r\right)={\displaystyle \frac{{\displaystyle {\int }_0^r{(sinhs/2)}^{2m}{(sinhs)}^{2\alpha +1}{(coshs)}^{2\beta +1}\mathrm{d}s}}{{\displaystyle m!{(\alpha +1)}_m{\int }_0^r{(sinhs)}^{2\alpha +1}{(coshs)}^{2\beta +1}\mathrm{d}s}}}.$$

(132)

Moreover,

$${\mathcal{B}}_r^{\alpha ,\beta }f=f+{\displaystyle \frac{r^2}{4(\alpha +2)}}{\mathcal{L}}_{\alpha ,\beta }f+O\left(r^4\right)(r\downarrow 0)$$

(133)

in the band-limited Schwartz topology. More precisely, the $O\left(r^4\right)$ remainder is uniform in every defining seminorm of ${\mathcal{B}}_R^{\alpha ,\beta }$; for each continuous seminorm p on that Fréchet space and each fixed $f\in {\mathcal{B}}_R^{\alpha ,\beta }$ there exist $C_{p,f}>0$ and $r_{p,f}>0$ such that

$$p\left({\mathcal{B}}_r^{\alpha ,\beta }f-f-{\displaystyle \frac{r^2}{4(\alpha +2)}}{\mathcal{L}}_{\alpha ,\beta }f\right)\le C_{p,f}{r}^4,0<r<r_{p,f}.$$

More generally, on every bounded subset $E\subset {\mathcal{B}}_R^{\alpha ,\beta }$ one may choose $C_{p,E}$ and $r_{p,E}$ uniformly for $f\in E$.

Proof. 

Insert (80) into (130). Uniform convergence on compact subintervals of $|sinhs|<1$ justifies termwise integration and yields (131). Since

$${(sinhs)}^{2\alpha +1}{(coshs)}^{2\beta +1}=s^{2\alpha +1}(1+O\left(s^2\right)),{\left({\displaystyle \frac{sinhs}{2}}\right)}^2={\displaystyle \frac{s^2}{4}}+O\left(s^4\right),$$

one obtains

$$c_1^{\alpha ,\beta }\left(r\right)={\displaystyle \frac{r^2}{4(\alpha +2)}}+O\left(r^4\right),$$

while $c_m^{\alpha ,\beta }\left(r\right)=O\left(r^{2m}\right)$ for $m\ge 2$, proving (133). □

7.2. A Brief Spectral-Filter Interpretation

In this section we record only the following operator-theoretic observation. This subsection is, therefore, an analytic spectral-filter interpretation, not an empirical machine-learning application. The operator $Q_{N,r}^{\alpha ,\beta }$ is the degree-N Pizzetti approximation to the exact mean at radius r, ${\epsilon }_N(R,r)$ is its explicit uniform spectral-band error bound, and ${\mathcal{H}}_{\theta ,N}$ is only a finite linear combination of such polynomial filters over several radii. No training procedure, discretization theorem, or empirical performance claim is asserted. To formulate operator norms precisely, let

$${\mathcal{H}}_R^{\alpha ,\beta }={\overline{{\mathcal{B}}_R^{\alpha ,\beta }}}^{L_{\alpha ,\beta }^2}.$$

(134)

Equivalently, in the Plancherel representation (34), ${\mathcal{H}}_R^{\alpha ,\beta }$ is the closed subspace on which both transform components ${\mathcal{F}}_{\alpha ,\beta }f$ and ${\mathcal{F}}_{\alpha ,\beta }\left(Rf\right)$ vanish almost everywhere for $\lambda >R$. Bounded spectral multipliers supported on $[0,R]$, therefore, extend uniquely to bounded operators on ${\mathcal{H}}_R^{\alpha ,\beta }$. To keep the notation transparent, $Q_{N,r}^{\alpha ,\beta }$ denotes the finite degree-N Pizzetti approximation, while

$$E_{N,r}^{\alpha ,\beta }:={\mathcal{M}}_r^{\alpha ,\beta }-Q_{N,r}^{\alpha ,\beta }$$

denotes the corresponding error operator. The scalar ${\epsilon }_N(R,r)$ below is an upper bound for the operator norm of $E_{N,r}^{\alpha ,\beta }$ on the fixed spectral band, and ${\mathcal{H}}_{\theta ,N}$ denotes only the finite multi-radius linear combination introduced at the end of the subsection. For $N\ge 0$ and $|sinhr|<1$, define

$$Q_{N,r}^{\alpha ,\beta }={\displaystyle \sum _{m=0}^N}{\displaystyle \frac{1}{m!{(\alpha +1)}_m}}{\left({\displaystyle \frac{sinhr}{2}}\right)}^{2m}{\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right).$$

(135)

On the spectral band ${\mathcal{B}}_R^{\alpha ,\beta }$, Theorem 2 and the Plancherel formula give

$${∥E_{N,r}^{\alpha ,\beta }f∥}_{L_{\alpha ,\beta }^2}\le {\epsilon }_N(R,r){\parallel f\parallel }_{L_{\alpha ,\beta }^2},$$

(136)

where

$${\epsilon }_N(R,r)={\displaystyle \sum _{m=N+1}^{\infty }}{\displaystyle \frac{{|sinhr|}^{2m}}{4^{m}m!{(\alpha +1)}_m}}{\displaystyle \prod _{j=0}^{m-1}}\left[R^2+{(\rho +2j)}^2\right],{\epsilon }_N(R,r)\to 0.$$

(137)

If $e_m(R,r)$ denotes the mth summand of the positive series in (137), then

$${\displaystyle \frac{e_{m+1}(R,r)}{e_m(R,r)}}={\displaystyle \frac{{|sinhr|}^2}{4}}{\displaystyle \frac{R^2+{(\rho +2m)}^2}{(m+1)(m+\alpha +1)}}⟶{|sinhr|}^2<1.$$

(138)

Hence the error series converges by the ratio test. More quantitatively, if q satisfies ${|sinhr|}^2<q<1$, then for all sufficiently large m the ratio in (138) is at most q, and consequently

$${\epsilon }_N(R,r)\le {\displaystyle \frac{e_{N+1}(R,r)}{1-q}}$$

for all sufficiently large N. Moreover, the Plancherel formula gives the genuine operator-norm estimate

$${∥E_{N,r}^{\alpha ,\beta }∥}_{\mathcal{L}\left({\mathcal{H}}_R^{\alpha ,\beta }\right)}\le {\epsilon }_N(R,r).$$

(139)

Thus, the phrase “operator-norm convergence on a fixed spectral band” refers precisely to the Hilbert subspace (134), not to the Fréchet space ${\mathcal{B}}_R^{\alpha ,\beta }$ itself. Thus, finite Pizzetti truncations are explicit reflection-equivariant polynomial approximants to the nonlocal mean on compact spectral bands. This is mathematically analogous to the use of polynomial spectral filters in geometric learning [13,14], but no numerical experiment, computational-complexity advantage, or empirical learning claim is made here.

For comparison with standard polynomial filters, if $r_1,\dots ,r_Q$ are fixed and $\theta =({\theta }_1,\dots ,{\theta }_Q)$, we may form only the linear filter bank

$${\mathcal{H}}_{\theta ,N}={\displaystyle \sum _{q=1}^Q}{\theta }_q{Q}_{N,r_q}^{\alpha ,\beta }={\displaystyle \sum _{m=0}^N}{b}_m\left(\theta \right){\mathcal{P}}_m^{\alpha ,\beta }\left({\mathcal{L}}_{\alpha ,\beta }\right),$$

where

$$b_m\left(\theta \right)={\displaystyle \frac{1}{m!{(\alpha +1)}_m}}{\displaystyle \sum _{q=1}^Q}{\theta }_q{\left({\displaystyle \frac{sinh{r}_q}{2}}\right)}^{2m}.$$

After discretization of ${\mathcal{L}}_{\alpha ,\beta }$ by a sparse matrix $L_h$, evaluating a degree-N Pizzetti filter by the shifted recurrence requires N applications of $L_h$; its asymptotic cost is, therefore, $O\left(N\mathrm{nnz}\left(L_h\right)\right)$, the same order as a degree-N Chebyshev graph filter. The distinction is approximation-theoretic. Chebyshev filters use a generic polynomial basis (after spectral rescaling) and are attractive for near-minimax uniform approximation, whereas the Pizzetti basis is tailored to the exact Jacobi multiplier and yields the explicit geometry-dependent tail (137). We do not claim a lower computational cost or better generic approximation rate. Establishing numerical stability for a particular discretization of the singular reflection–differential operator is a separate problem. We regard such numerical questions as outside the scope of the present harmonic-analysis paper.

8. Conclusions

We have developed a Pizzetti calculus for the rank-one Opdam–Cherednik Laplacian in which the classical Jacobi hypergeometric expansion is realized as a convergent operator identity in the nonsymmetric reflection setting. The decisive step is the Weyl symmetrization of the generalized translation; it replaces the nonsymmetric multiplier by the Jacobi multiplier while retaining the full reflection–differential operator in the spatial variable. The associated Darboux equation then produces the shifted polynomial factors intrinsically.

The infinite expansion is justified on compact spectral bands in the Opdam–Cherednik Paley–Wiener topology. Its consequences include local mean-value and finite shifted-polyharmonic principles, a normalized ball-mean formula, and a quantitative two-radius estimate. The curvature contraction is formulated in $C_{\mathrm{loc}}^{\infty }$ for every fixed polynomial order, while the full rational Dunkl series is established independently from the rational multiplier. The even rank-one symmetric-space cases are included as consistency specializations of the nonsymmetric theory. A final brief remark notes the polynomial spectral-filter interpretation of the truncations, without making an empirical machine-learning claim. We stress that the Pizzetti formula proved here is a local expansion in the radius variable, valid in the hypergeometric power-series range $|sinhr|<1$ on each fixed compact spectral band. This local restriction does not limit the definition of the symmetrized mean itself, which exists for a broader set of real radii. The scope of the contribution should also be kept explicit. The Jacobi hypergeometric expansion, the classical Darboux and Berenstein–Zalcman mechanisms, and the symmetric-space special cases are part of the established literature. The present results concern their rigorous realization in the nonsymmetric rank-one Opdam–Cherednik calculus and the convergence statements needed for that realization. Moreover, the local mean-value and shifted-polyharmonic equivalences are conditional statements inside the locally Pizzetti-admissible class of Definition 1; they are not asserted as characterizations of arbitrary smooth functions. A final editorial pass has also been made to keep the terminology, reflection notation, local radius condition, convergence topologies, parameter conventions, equation references, and spectral-filter notation uniform throughout the manuscript. Higher-rank Cherednik systems and global spectral-synthesis questions are natural directions for future work.