A Study of the Generalized Gabor Transform with Applications to Reproducing Kernel Theory

Abstract

The aim of this paper is to establish an inversion and Calderón formulas for the generalized Gabor transform associated with a class of Sturm–Liouville operators. We also investigate several problems related to reproducing kernel theory for this transform. In particular, we study the concept of Tikhonov regularization and the extremal functions associated with the new generalized Gabor transform.

1. Introduction

The Fourier transform is a cornerstone of mathematical analysis, with broad applications in signal processing, physics, and engineering. While effective for stationary signals, many real-world signals are non-stationary, requiring time–frequency methods. The Gabor transform, or short-time Fourier transform (STFT), decomposes signals into time- and frequency-shifted windows. It finds applications in harmonic analysis, sampling theory, quantum mechanics, geophysics, medicine, and signal/image processing [1,2,3,4,5,6,7,8]. Its theory has also been studied in abstract settings, such as hypergroups [9], locally compact Abelian/non-Abelian groups [10,11,12], and Gelfand pairs [13].

A natural generalization arises via Sturm–Liouville operators on ${\mathbb{R}}_{+}$:

$$L_A:={\displaystyle \frac{d^2}{d{x}^2}}+{\displaystyle \frac{A^{\prime }\left(x\right)}{A\left(x\right)}}{\displaystyle \frac{d}{dx}},A\left(x\right)=x^{2\alpha +1}B\left(x\right),\alpha >-{\displaystyle \frac{1}{2}},$$

(1)

where $B\left(x\right)$ is a positive, $C^{\infty }$, and even function. We assume:

• A is increasing with ${\displaystyle \underset{x\to \infty }{lim}A\left(x\right)=\infty }$;
• $A^{\prime }/A$ is decreasing with ${\displaystyle \underset{x\to \infty }{lim}{A}^{\prime }/A=2\rho \ge 0}$;
• for some $\delta >0$ and bounded $C^{\infty }$ function D,

  $${\displaystyle \frac{A^{\prime }\left(x\right)}{A\left(x\right)}}=2\rho +e^{-\delta x}D\left(x\right),x\in [x_0,\infty ),x_0>0.$$

  (2)

Under these conditions, $L_A$ induces the Chébli–Trimèche hypergroup, with eigenfunctions acting as hypergroup characters. This allows harmonic analysis analogous to the classical case. Recent results include the generalized Fourier transform, g-function [14], transform ranges [15,16], uncertainty principles [17,18], maximal functions [19], variation-diminishing kernels [20], spectral theorems and convolution products [21], Paley–Wiener theorems and transmutation operators [22,23], heat equation [24], Delsarte’s formula [25], and wavelet transforms [26].

Within this framework, in this paper, we continue the study of the generalized Gabor transform (GGT) in the Chébli–Trimèche/hypergroup frame, initiated in [27]. Our objectives are:

• Establish inversion and Calderón-type formulas for the GGT;
• Develop the reproducing kernel theory, including Moore–Penrose inverses, Tikhonov regularization, and extremal functions.

The theory of reproducing kernels, developed by Aronszajn [28], extended by Schwartz [29], and formalized by Saitoh [30], is fundamental for these developments. In particular, for any even $\phi \in L_{{\nu }_{A,b}}^2\left(\mathbb{R}\right)$ and $s>0$, the extremal function

$$f_{r,g}^{*}=\underset{f\in W_{A,m}^s\left(\mathbb{R}\right)}{min}\left\{{r\parallel f\parallel }_{W_{A,m}^s}^2+{\parallel g-{\mathfrak{G}}_{\phi }^{A,m}\left(f\right)\parallel }_{L_{{\mu }_{A,b}}^2}^2\right\}$$

exists and is explicitly given by

$$f_{r,g}^{*}\left(x\right)={\int }_{{\mathbb{R}}^2}g(t,y){\mathfrak{S}}_{r,\phi }^{A,m}(t,x,y)d{\mu }_{A,b}(t,y),$$

with kernel ${\mathfrak{S}}_{r,\phi }^{A,m}$ as defined in the main text. Here $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ denotes the generalized Paley–Wiener space given in Equation (34).

The remainder of the paper is organized as follows. In Section 2, we recall harmonic analysis results for the generalized canonical Fourier transform [31]. In Section 3, we revisit the GGT and present new results. In Section 4, we develop the reproducing kernel theory for the GGT setting and discuss applications, including Tikhonov regularization and extremal functions.

2. Preliminaries

This section introduces the harmonic analysis associated with the Lions operator $L_A$. The main references are [21,23]. First let us fix some notations:

• For $p\in [1,\infty ]$, the conjugate exponent is $p^{\prime }$.
• $L_{{\gamma }_A}^p\left({\mathbb{R}}_{+}\right)$, $1\le p\le \infty$, is the space of functions f on ${\mathbb{R}}_{+}$ with

  $$\begin{array}{cc}\hfill {\parallel f\parallel }_{L_{{\gamma }_A}^p}& :={\left({\int }_{{\mathbb{R}}_{+}}{\left|f\right|}^{p}d{\gamma }_A\right)}^{1/p}<\infty ,1\le p<\infty ,\hfill \\ \hfill {\parallel f\parallel }_{L_{{\gamma }_A}^{\infty }}& {\displaystyle :=\mathrm{ess}{\sup }_{t\in {\mathbb{R}}_{+}}\left|f\left(t\right)\right|<\infty ,}\hfill \end{array}$$

  where

  $$d{\gamma }_A\left(t\right):=A\left(t\right)dt.$$
  For $p=2$, this space is equipped with the scalar product

  $${\langle f,g\rangle }_{L_{{\gamma }_A}^2}:={\int }_{{\mathbb{R}}_{+}}f\left(t\right)\overline{g\left(t\right)}d{\gamma }_A\left(t\right).$$
• $L_{{\nu }_A}^p\left({\mathbb{R}}_{+}\right)$, $1\le p\le \infty$, is defined similarly with measure

  $$d{\nu }_A\left(\lambda \right):={\displaystyle \frac{d\lambda }{2\pi |c_A{\left(\lambda \right)|}^2}},$$

  where $c_A$ is the Harish-Chandra function related to $L_A$ [21].
• $C_{c,e}\left(\mathbb{R}\right)$: Space of even, continuous, compactly supported functions.
• $C_b\left({\mathbb{R}}_{+}\right)$: Space of bounded continuous functions on ${\mathbb{R}}_{+}$.
• ${\mathcal{S}}_e\left(\mathbb{R}\right)$: Schwartz space of rapidly decreasing even functions.
• Generalized Schwartz space ${\mathcal{S}}^2\left(\mathbb{R}\right):={(coshx)}^{-2\varrho }{\mathcal{S}}_e\left(\mathbb{R}\right)$, with seminorms

  $$P_{m,n}\left(f\right):=\underset{x\in \mathbb{R},0\le j\le n}{sup}{(coshx)}^{2\varrho }{(1+x^2)}^m\left|{\displaystyle \frac{d^j}{d{x}^j}}f\left(x\right)\right|<\infty ,$$

  where ${\displaystyle 2\varrho :=\underset{x\to \infty }{lim}{A}^{\prime }/A}$.
• $SL(2,\mathbb{R})$: The group of $2\times 2$ real matrices with determinant one.

The Lions operator $L_A$ satisfies the following properties:

1.

For every $\lambda \in \mathbb{C}$, the equation

$$L_{A}u=-({\lambda }^2+{\varrho }^2)u,u\left(0\right)=1,u^{\prime }\left(0\right)=0,$$

admits a unique $C^{\infty }$ solution ${\varphi }_{\lambda }$ on ${\mathbb{R}}_{+}$.

2.

For all $t\in {\mathbb{R}}_{+}$, the function $\lambda \mapsto {\varphi }_{\lambda }\left(t\right)$ is analytic.

3.

For all $t,\lambda \in {\mathbb{R}}_{+}$,

$$|{\varphi }_{\lambda }\left(t\right)|\le 1.$$

(3)

4.

For every $\lambda \in \mathbb{C}$, $t\in {\mathbb{R}}_{+}$ and $n\in \mathbb{N}$,

$$\left|{\displaystyle \frac{d^n}{d{\lambda }^n}}{\varphi }_{\lambda }\left(t\right)\right|\le {\left|t\right|}^n{e}^{|\mathrm{Im}\lambda ||t|}.$$

Remark 1.

1.

If $A\left(t\right)=t^{2\alpha +1}$, $\alpha >-{\displaystyle \frac{1}{2}}$, then $L_A$ reduces to the Bessel operator, with

$${\varphi }_{\lambda }\left(z\right)=j_{\alpha }\left(z\right)=\Gamma (\alpha +1)\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{k!\Gamma (k+\alpha +1)}}{\left({\displaystyle \frac{z}{2}}\right)}^{2k}.$$

2.

If $A\left(t\right)={(sinht)}^{2\alpha +1}{(cosht)}^{2\beta +1}$, $\alpha \ge \beta \ge -{\displaystyle \frac{1}{2}}$, $\alpha \ne -{\displaystyle \frac{1}{2}}$, then $L_A$ reduces to the Jacobi operator, with

$${\varphi }_{\lambda }\left(x\right)={}_{2}F_1\left({\displaystyle \frac{\rho +\lambda }{2}},{\displaystyle \frac{\rho -\lambda }{2}},\alpha +1,-{(sinhx)}^2\right),$$

where ${}_{2}F_1$ is the Gauss hypergeometric function.

2.1. Generalized Fourier Transform

For $f\in C_c\left({\mathbb{R}}_{+}\right)$, the Fourier transform associated with $L_A$ is

$${\mathcal{F}}_A\left(f\right)\left(\lambda \right):={\int }_{{\mathbb{R}}_{+}}f\left(x\right){\varphi }_{\lambda }\left(x\right)d{\gamma }_A\left(x\right),$$

with inverse

$${\mathcal{F}}_A^{-1}\left(g\right)\left(x\right)={\int }_{{\mathbb{R}}_{+}}g\left(\lambda \right){\varphi }_{\lambda }\left(x\right)d{\nu }_A\left(\lambda \right).$$

(4)

Definition 1.

For $f\in C_b\left({\mathbb{R}}_{+}\right)$ and $x\in {\mathbb{R}}_{+}$, the generalized translation operator ${\tau }_x^A$ is

$${\tau }_x^{A}f\left(y\right):={\int }_{{\mathbb{R}}_{+}}f\left(z\right)d{\mu }_{x,y}^A\left(z\right),$$

where $d{\mu }_{x,y}^A$ is supported on $\left[\right|x-y|,x+y]$.

Then, we have the following results [21,23]:

1.

If $f\in L_{\mathrm{loc}}^1\left(d{\gamma }_A\right)$, then ${\tau }_x^{A}f\left(y\right)={\tau }_y^{A}f\left(x\right)$ and ${\tau }_0^{A}f=f$.

2.

If $f\in L_{{\gamma }_A}^p\left({\mathbb{R}}_{+}\right)$, $1\le p\le \infty$, then

$$\parallel {\tau }_x^A{f\parallel }_{L_{{\gamma }_A}^p}\le {\parallel f\parallel }_{L_{{\gamma }_A}^p}.$$

3.

If $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$, then

$${\mathcal{F}}_A\left({\tau }_x^{A}f\right)\left(\lambda \right)={\varphi }_{\lambda }\left(x\right){\mathcal{F}}_A\left(f\right)\left(\lambda \right).$$

(5)

4.

For $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$ such that $f\ge 0$, then ${\tau }_x^{A}f\ge 0$.

5.

If $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$, then

$${\int }_{{\mathbb{R}}_{+}}{\tau }_x^{A}f\left(y\right)d{\gamma }_A\left(y\right)={\int }_{{\mathbb{R}}_{+}}f\left(y\right)d{\gamma }_A\left(y\right).$$

2.2. Generalized Canonical Fourier Transform

Let $m=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2\left(\mathbb{R}\right)$, with $b\ne 0$.

For $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$, the generalized canonical Fourier transform [31] is

$${\mathfrak{F}}_A^m\left(f\right)\left(\lambda \right)={\int }_{{\mathbb{R}}_{+}}f\left(x\right)K_A^m(x,\lambda )d{\gamma }_A\left(x\right),$$

where

$$K_A^m(x,\lambda )={\mathrm{e}}^{{\displaystyle \frac{i}{2}}({\displaystyle \frac{d}{b}}{x}^2+{\displaystyle \frac{a}{b}}{\lambda }^2)}{\varphi }_{\lambda /b}\left(x\right).$$

From (3), we derive

$$|K_A^m(x,\lambda )|\le 1.$$

(6)

Special cases: The generalized canonical Fourier transform recover the Fresnel transform for $m=\left(\begin{array}{cc}1& \tau \\ 0& 1\end{array}\right)$, $\tau \in {\mathbb{R}}_{+}$; the fractional Fourier transform for $m=\left(\begin{array}{cc}cosh\left(\tau \right)& sinh\left(\tau \right)\\ sinh\left(\tau \right)& cosh\left(\tau \right)\end{array}\right)$, $\tau \in {\mathbb{R}}_{+}$; and the generalized Fourier transform ${\mathcal{F}}_A$ for $m:=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. (For more details, see [31].)

The differential operator ${\Delta }_A^m$ of the Sturm–Liouville type defined by

$$\begin{array}{ccc}\hfill {\Delta }_A^{m}f& =& {\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}}f+\left({\displaystyle \frac{A^{\prime }\left(x\right)}{A\left(x\right)}}-2i{\displaystyle \frac{d}{b}}x\right){\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}f-{\displaystyle \frac{d}{b}}\left(i+{\displaystyle \frac{d}{b}}{x}^2+ix{\displaystyle \frac{A^{\prime }\left(x\right)}{A\left(x\right)}}\right)f,\hfill \end{array}$$

has the following properties:

• ${\Delta }_A^m$ and $L_A$ are connected by

  $$M_{-{\displaystyle \frac{d}{b}}}\circ {\Delta }_A^m\circ M_{{\displaystyle \frac{d}{b}}}=L_A,$$

  where $M_s$ is defined by $M_{s}f\left(x\right)={\mathrm{e}}^{{\displaystyle \frac{is}{2}}{x}^2}f\left(x\right)$.
• The kernel $K_A^m(x,\lambda )$ is the unique solution of

  $$\left\{\begin{array}{c}{\Delta }_A^m{K}_A^m(·,\lambda )=-({\displaystyle \frac{{\lambda }^2}{b^2}}+{\varrho }^2)K_A^m(·,\lambda )\hfill \\ \\ K_A^m(0,\lambda )={\mathrm{e}}^{{\displaystyle \frac{i}{2}}{\displaystyle \frac{a}{b}}{\lambda }^2},{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}{K}_A^m(0,\lambda )=0.\hfill \end{array}\right.$$
• If $f,g\in {\mathcal{S}}^2\left(\mathbb{R}\right)$, then

  $${\int }_{{\mathbb{R}}_{+}}{\Delta }_A^{m}f\overline{g}d{\gamma }_A={\int }_{{\mathbb{R}}_{+}}f\overline{{\Delta }_A^{m}g}d{\gamma }_A.$$

We denoted by $L_{{\nu }_{A,b}}^p\left({\mathbb{R}}_{+}\right)$ the space of functions, with

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_{{\nu }_{A,b}}^p}& =& {\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|f\right|}^{p}d{\nu }_{A,b}}\right)}^{1/p}<\infty ,p\in [1,\infty ),\hfill \\ \hfill {\parallel f\parallel }_{L_{{\nu }_{A,b}}^{\infty }}& =& {\displaystyle \mathrm{ess}\underset{\xi \in {\mathbb{R}}_{+}}{sup}\left|f\left(\xi \right)\right|<\infty ,}\hfill \end{array}$$

where

$$d{\nu }_{A,b}\left(\xi \right)=d{\nu }_A(\xi /b).$$

Then we have:

• Riemann–Lebesgue-type lemma: For every $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$, we have ${\mathfrak{F}}_A^m\left(f\right)\in C_0\left({\mathbb{R}}_{+}\right)$ and

  $$\parallel {\mathfrak{F}}_A^m{\left(f\right)\parallel }_{\infty }\le {\parallel f\parallel }_{L_{{\gamma }_A}^1}.$$
• Inversion-type formula: If $f\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$ such that ${\mathfrak{F}}_A^m\left(f\right)\in L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$, then

  $${\displaystyle f\left(x\right)={\int }_{{\mathbb{R}}_{+}}{\mathfrak{F}}_A^m\left(f\right)\left(\xi \right)\overline{K_A^m(x,\xi )}d{\nu }_A\left(\xi /b\right),x\in {\mathbb{R}}_{+}.}$$
• Parseval-type formulas:

  (a)

  For $f,g\in {\mathcal{S}}^2\left(\mathbb{R}\right)$,

  $${\displaystyle {\int }_{{\mathbb{R}}_{+}}f\left(t\right)\overline{g\left(t\right)}d{\gamma }_A\left(t\right)={\int }_{{\mathbb{R}}_{+}}{\mathfrak{F}}_A^m\left(f\right)\left(\xi \right)\overline{{\mathcal{F}}_A^m\left(g\right)\left(\xi \right)}d{\nu }_A\left(\xi /b\right).}$$

  (7)

  In particular, we have the following Plancherel-type formula:

  $${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|f\left(t\right)\right|}^{2}d{\gamma }_A\left(t\right)={\int }_{{\mathbb{R}}_{+}}{\left|{\mathfrak{F}}_A^m\left(f\right)\left(\xi \right)\right|}^{2}d{\nu }_A\left(\xi /b\right).}$$

  (8)

  (b)

  ${\mathfrak{F}}_A^m$ extends to an isometric isomorphism from $L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$ onto $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$. Moreover, for all $f\in L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$,

  $${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|f\left(t\right)\right|}^{2}d{\gamma }_A\left(t\right)={\int }_{{\mathbb{R}}_{+}}{\left|{\mathfrak{F}}_A^m\left(f\right)\left(\xi \right)\right|}^{2}d{\nu }_A\left(\xi /b\right).}$$

  (9)

3. Generalized Gabor Transform (GGT)

This section is devoted to proving new results for the GGT. First, we begin by recalling the main results concerning this transform established in [27]. We denoted by $L_{{\mu }_{A,b}}^p\left({\mathbb{R}}_{+}^2\right)$, $1\le p\le \infty$, the space of functions f on ${\mathbb{R}}_{+}^2$ satisfying

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L_{{\mu }_{A,b}}^p}& =& {\left({\displaystyle {\int }_{{\mathbb{R}}_{+}^2}{\left|f(x,\lambda )\right|}^{p}d{\mu }_{A,b}(x,\lambda )}\right)}^{1/p}<\infty ,1\le p<\infty ,\hfill \\ \hfill {\parallel f\parallel }_{L_{{\mu }_{A,b}}^{\infty }}& =& \underset{(x,\lambda )\in {\mathbb{R}}_{+}^2}{\mathrm{ess}sup}\left|f(x,\lambda )\right|<\infty ,p=\infty ,\hfill \end{array}$$

where ${\mathbb{R}}_{+}^2:={\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ and $d{\mu }_{A,b}(x,\lambda ):=d{\nu }_{A,b}\left(x\right)d{\gamma }_A\left(\lambda \right)$. Then the generalized convolution product is defined on ${\mathcal{S}}_e\left(\mathbb{R}\right)$ by

$$\begin{array}{ccc}\hfill \forall \xi \in {\mathbb{R}}_{+},u\underset{A,m}{\ast }v\left(\xi \right)& =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(x\right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left(v\right)\left(x\right)K_A^m(x,\xi )d{\gamma }_A\left(x\right)}\hfill \\ & =& {\mathfrak{F}}_m^A\left({\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left(v\right)\right)\left(\xi \right),u,v\in {\mathcal{S}}_e\left(\mathbb{R}\right).\hfill \end{array}$$

We have the following properties:

• If $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $v\in L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$, then $u\underset{A,m}{\ast }v$ is in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, with

  $${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\underset{A,m}{\ast }v\right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left(v\right).$$

  (10)
• If $u,v\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, then $u\underset{A,m}{\ast }v$ belongs to $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ if and only if ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left(v\right)\in L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$, such that

  $${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\underset{A,m}{\ast }v\right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left(v\right).$$

  (11)
• If $u,v\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, then

  $${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|u\underset{A,m}{\ast }v\left(\xi \right)\right|}^{2}d{\nu }_{A,b}\left(\xi \right)={\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\underset{A,m}{\ast }v\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(\xi \right).}$$

  (12)

The modulation of $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ by the real number s is given by

$${\phi }_s^{A,m}:={\mathfrak{F}}_m^A\left(\sqrt{{\tau }_s^A\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)}\right).$$

(13)

Then

$$\parallel {\phi }_s^{A,m}{\parallel }_{L_{{\nu }_{A,b}}^2}={\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}.$$

(14)

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we defined the generalized Gabor transform (GGT), by

$${\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(t,s):=u\underset{A,m}{\ast }{\phi }_s^{A,m}\left(t\right)={\langle u,{\phi }_{t,s}^{A,m}\rangle }_{L_{{\nu }_{A,b}}^2},$$

(15)

where ${\phi }_{t,s}^{A,m}$ is defined by

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{t,s}^{A,m}\right)\left(\xi \right):=\overline{{\left({\mathfrak{F}}_m^A\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)K_A^m(s,\xi )}.$$

(16)

Proposition 1.

For $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have

$${∥{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)∥}_{L_{{\mu }_{A,b}}^{\infty }}\le {\parallel u\parallel }_{L_{{\nu }_{A,b}}^2}{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}.$$

Proposition 2

(Plancherel-type formula). For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have for every $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$,

$${∥{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)∥}_{L_{{\mu }_{A,b}}^2}={\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}{\parallel u\parallel }_{L_{{\nu }_{A,b}}^2}.$$

As in the classical setting, the GGT preserves the orthogonality property. More precisely, we have the following result.

Corollary 1.

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have for every $u,v\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right),$

$${\displaystyle {\displaystyle {\int }_{{\mathbb{R}}_{+}^2}{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(y,s)\overline{{\mathfrak{G}}_{\phi }^{A,m}\left(v\right)(y,s)}d{\mu }_{A,b}(y,s)}={\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2{\int }_{{\mathbb{R}}_{+}}u\left(x\right)\overline{v\left(x\right)}d{\nu }_{A,b}\left(x\right).}$$

(17)

Proposition 3.

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have for every $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $p\in [2,\infty ),$

$$\begin{array}{c}\hfill {∥{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)∥}_{L_{{\mu }_{A,b}}^p}⩽{\parallel u\parallel }_{L_{{\nu }_{A,b}}^2}{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}.\end{array}$$

A straightforward computation yields the following results.

Proposition 4.

Let $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$ be a non-trivial function. Then

1. 

For every $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$,

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\sqrt{{\tau }_s^A\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)\left(\xi \right)}.$$

2. 

${\mathfrak{G}}_{\phi }^{A,m}\left(L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\right)$ is a reproducing kernel Hilbert space (RKHS) in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ with kernel

$${\displaystyle {\mathfrak{K}}_{\phi }^{A,m}(x^{\prime },s^{\prime };x,s):=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}{\phi }_{x^{\prime },s^{\prime }}^{A,m}\left(y\right)\overline{{\phi }_{x,s}^{A,m}\left(y\right)}d{\nu }_{A,b}\left(y\right),}$$

such that, for all $(x^{\prime },s^{\prime }),(x,s)\in {\mathbb{R}}_{+}^2$,

$$\left|{\mathfrak{K}}_{\phi }^{A,m}(x^{\prime },s^{\prime };x,s)\right|\le 1.$$

We are now ready to establish the following inversion formula for the GGT.

Theorem 1

(Inversion formula). For a nonzero function $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have for every $u\in L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ such that ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right),$

$$\begin{array}{c}\hfill {\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(x,s)\overline{{\phi }_{y,s}^{A,m}\left(x\right)}d{\nu }_{A,b}\left(x\right)}\right)d{\gamma }_A\left(s\right),a.e.,}\end{array}$$

where ${\phi }_{y,s}^{A,m}$ is defined by (16).

In order to prove the last result, we require the following lemma.

Lemma 1

($L_{{\nu }_{A,b}}^2$-Inversion formula). For a nonzero function $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, we have for every $u\in L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right),$

$${\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}\underset{n\to \infty }{lim}{\int }_{-n}^n{\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)d{\gamma }_A\left(\xi \right)d{\gamma }_A\left(s\right),}$$

where the limit is in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right).$

Proof. 

Let u be in $L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$. Then for every $\xi \in {\mathbb{R}}_{+}$,

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right).$$

(18)

Moreover, by (13), we have

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)=\sqrt{{\tau }_s^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\xi \right)}.$$

(19)

Then, since $|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right),$ it follows from (5) that the function

$$\lambda \mapsto {\mathcal{F}}_A({\tau }_{\xi }^A|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)$$

is continuous on ${\mathbb{R}}_{+}$, such that

$${\mathcal{F}}_A\left({\tau }_{\xi }^A\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)\right)\left(\lambda \right)={\varphi }_{\lambda }\left(\xi \right){\mathfrak{F}}_A^m\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)\left(\lambda \right).$$

For $\lambda =0$, we derive by (9),

$${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\tau }_s^A\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)\left(\xi \right)d{\gamma }_A\left(s\right)={\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2.}$$

(20)

Thus, using (19) and (20), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|\left({\mathfrak{F}}_A^m{)}^{-1}({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right)={\int }_{{\mathbb{R}}_{+}}{\tau }_s^A\left({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2\right)\left(\xi \right)d{\gamma }_A\left(s\right)={\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2.}$$

(21)

It follows that for every $\xi \in {\mathbb{R}}_{+}$, the function $s\mapsto {\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)$ belongs to $L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$ and the function $s\mapsto {\tau }_s^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\xi \right)$ is in $L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$.

Then involving (18) and (19), we get

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)d{\gamma }_A\left(s\right)}\hfill \\ & =& {\displaystyle {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\int }_{{\mathbb{R}}_{+}}{\tau }_s^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\xi \right)d{\gamma }_A\left(s\right)}\hfill \\ & =& {\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right).\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill {\displaystyle {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)d{\gamma }_A\left(s\right).}\end{array}$$

Thus, using this relation and (16),

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_0^n\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(s\right)}\right)d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_0^n{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)K_A^m(y,\xi )d{\gamma }_A\left(\xi \right).}\hfill \end{array}$$

(22)

As u is in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right),$ we have in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right),$

$${\displaystyle \underset{n\to \infty }{lim}{\int }_0^n{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)K_A^m(y,\xi )d{\gamma }_A\left(\xi \right)=u\left(y\right),y\in {\mathbb{R}}_{+}}$$

Hence, by (22), we obtain in $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$,

$${\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}\underset{n\to \infty }{lim}{\int }_0^n\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(s\right)}\right)d{\gamma }_A\left(\xi \right),}$$

(23)

as desired.   □

Proof of Theorem 1.

From (23), we derive that for a.e. $y\in {\mathbb{R}}_{+}$,

$${\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}\underset{n\to \infty }{lim}{\int }_0^n\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(s\right)}\right)d{\gamma }_A\left(\xi \right).}$$

Moreover, by (3), (18) and (21), we have

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_0^n{\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)\right|d{\gamma }_A\left(s\right)d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_0^n{\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)\right|d{\gamma }_A\left(s\right)d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_0^n{\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right||{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right){|}^2|K_A^m(y,\xi )|d{\gamma }_A\left(s\right)d{\gamma }_A\left(\xi \right)}\hfill \\ & \le & {\displaystyle {\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right|\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\tau }_{\xi }^A({\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\right|}^2)\left(s\right)d{\gamma }_A\left(s\right)}\right)d{\gamma }_A\left(\xi \right)}\hfill \\ & \le & \parallel {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\parallel }_{L_{{\nu }_{A,b}}^1}{\parallel {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\parallel }_{L_{{\gamma }_A}^2}^2\hfill \\ & \le & \parallel {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right){\parallel }_{L_{{\nu }_{A,b}}^1}{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2<\infty .\hfill \end{array}$$

(24)

Then for a.e. $y\in {\mathbb{R}}_{+}$,

$${\displaystyle u\left(y\right)=\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}_{+}}\left({\displaystyle {\int }_0^n{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(\xi \right)}\right)\frac{d{\gamma }_A\left(s\right)}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}.}$$

(25)

For $y\in {\mathbb{R}}_{+}$, let ${\left\{U_n\right\}}_n$ be the sequence defined by

$${\displaystyle U_n\left(s\right)={\int }_0^n{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(\xi \right),s\in {\mathbb{R}}_{+},}$$

which satisfies

$${\displaystyle \underset{n\to \infty }{lim}{U}_n\left(s\right)={\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(\xi \right),}$$

and

$${\displaystyle |U_n\left(s\right)|\le {\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)\right|d{\gamma }_A\left(\xi \right).}$$

Using the same arguments as in (24), we obtain that the function

$${\displaystyle s\mapsto {\int }_{{\mathbb{R}}_{+}}\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right){\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)\right|d{\gamma }_A\left(\xi \right)}$$

is integrable on ${\mathbb{R}}_{+}$ with respect to $d{\gamma }_A.$ Thus, by (25), we derive that, for almost every $y\in {\mathbb{R}}_{+}$,

$${\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s)\right)\left(\xi \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_{y,s}^{A,m}\right)\left(\xi \right)}d{\gamma }_A\left(\xi \right)}\right)d{\gamma }_A\left(s\right).}$$

Hence, by (7), we conclude that

$${\displaystyle u\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(x,s)\overline{{\phi }_{y,s}^{A,m}\left(x\right)}d{\nu }_{A,b}\left(x\right)}\right)d{\gamma }_A\left(s\right),a.e.y\in {\mathbb{R}}_{+},}$$

which completes the proof.    □

We conclude this subsection with the following result.

Theorem 2

(Calderón-type reproducing formula). For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ such that ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\in L_{{\gamma }_A}^{\infty }\left({\mathbb{R}}_{+}\right)$, we have for every $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $0<ϵ<\delta <\infty$ that the function

$${\displaystyle u^{ϵ,\delta }\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}{\int }_{{\mathbb{R}}_{+}}{\mathfrak{G}}_{\phi }^{A,m}\left(u\right)(·,s){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)d{\gamma }_A\left(s\right),y\in {\mathbb{R}}_{+},}$$

(26)

belongs to $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and satisfies

$$\underset{ϵ\to 0,\delta \to \infty }{lim}\left|\right|u^{ϵ,\delta }-u{\left|\right|}_{L_{{\nu }_{A,b}}^2}=0,$$

(27)

where

$$C(\epsilon ,\delta ):=\left\{x\in {\mathbb{R}}_{+}:\epsilon \le x\le \delta \right\}.$$

The proof of the last theorem requires the following lemmas.

Lemma 2.

If $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ such that ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\in L_{{\gamma }_A}^{\infty }\left({\mathbb{R}}_{+}\right)$, then the function

$${\displaystyle {\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right),\xi \in {\mathbb{R}}_{+},}$$

satisfies, for almost all $\xi \in {\mathbb{R}}_{+}$,

$$0<{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\le 1,$$

and

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)=1.$$

(28)

Proof. 

We have

$$\begin{array}{ccc}{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\hfill & =\hfill & {\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right)}\hfill \\ & \le \hfill & {\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right).}\hfill \end{array}$$

Involving Plancherel-type formula (8) and (14), we get

$$0<{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\le 1.$$

Moreover, using the assumption on the window function, we derive (28).    □

Lemma 3.

If $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ such that ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\in L_{{\gamma }_A}^{\infty }\left({\mathbb{R}}_{+}\right)$ and $u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, then the function $u^{ϵ,\delta }$ given by (26) belongs to $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and satisfies

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u^{ϵ,\delta }\right)\left(\xi \right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\mathfrak{K}}_{ϵ,\delta }\left(\xi \right),\xi \in {\mathbb{R}}_{+}.$$

Proof. 

By (15), $u^{ϵ,\delta }$ can be written as

$${\displaystyle u^{ϵ,\delta }\left(y\right)=\frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}\left(u{\ast }_{A,m}{\phi }_s^{A,m}\right){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)d{\gamma }_A\left(s\right).}$$

(29)

Then

$${\displaystyle |u^{ϵ,\delta }{\left(y\right)|}^2\le \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^4}\left({\displaystyle {\int }_{C(ϵ,\delta )}d{\gamma }_A\left(s\right)}\right){\int }_{C(ϵ,\delta )}{\left|\left(u{\ast }_{A,m}{\phi }_s^{A,m}\right){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)\right|}^{2}d{\gamma }_A\left(s\right).}$$

Therefore

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|u^{ϵ,\delta }\left(y\right)\right|}^{2}d{\nu }_{A,b}\left(y\right)}\hfill \\ & \le & {\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^4}\left({\displaystyle {\int }_{C(ϵ,\delta )}d{\gamma }_A\left(s\right)}\right){\int }_{C(ϵ,\delta )}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|\left(u{\ast }_{A,m}{\phi }_s^{A,m}\right){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)\right|}^{2}d{\nu }_{A,b}\left(y\right)}\right)d{\gamma }_A\left(s\right).}\hfill \end{array}$$

Thus by (9)–(11), we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|u^{ϵ,\delta }\left(y\right)\right|}^{2}d{\nu }_{A,b}\left(y\right)}\hfill \\ & \le & {\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^4}\left({\displaystyle {\int }_{C(ϵ,\delta )}d{\gamma }_A\left(s\right)}\right){\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right|}^2\left({\displaystyle {\int }_{C(ϵ,\delta )}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{4}d{\gamma }_A\left(s\right)}\right)d{\gamma }_A\left(\xi \right).}\hfill \end{array}$$

Moreover, since ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)\in L_{{\nu }_{A,b}}^{\infty }\left({\mathbb{R}}_{+}\right)$, then

$$\begin{array}{cc}\hfill {\int }_{C(ϵ,\delta )}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{4}d{\gamma }_A\left(s\right)& \le {\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^2{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right)\hfill \\ & \le {∥{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)∥}_{L_{{\gamma }_A}^{\infty }}^2{∥{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right)∥}_{L_{{\gamma }_A}^2}^2.\hfill \end{array}$$

Hence, by (9),

$${\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|u^{ϵ,\delta }\left(y\right)\right|}^{2}d{\gamma }_A\left(y\right)\le \left({\displaystyle {\int }_{C(ϵ,\delta )}d{\gamma }_A\left(s\right)}\right)\frac{\parallel {\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){\parallel }_{L_{{\gamma }_A}^{\infty }}^2{\parallel u\parallel }_{L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)}^2}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}<\infty ,}$$

which implies that $u^{ϵ,\delta }\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right).$

On the other hand, let $\psi$ in ${\mathcal{S}}^2\left(\mathbb{R}\right)$. Then, ${\mathfrak{F}}_A^m\left(\psi \right)\in {\mathcal{S}}_e\left(\mathbb{R}\right)$ and by (29),

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}_{+}}{u}^{ϵ,\delta }\left(y\right)\overline{{\mathfrak{F}}_A^m\left(\psi \right)\left(y\right)}d{\nu }_{A,b}\left(y\right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}\left({\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}\left(u{\ast }_{A,m}{\phi }_s^{A,m}\right){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)d{\gamma }_A\left(s\right)}\right)\overline{{\mathfrak{F}}_A^m\left(\psi \right)\left(y\right)}d{\nu }_{A,b}\left(y\right).}\hfill \end{array}$$

(30)

By proceeding as above, Equation (30) is equal to

$${\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}\left(u{\ast }_{A,m}{\phi }_s^{A,m}\right){\ast }_{A,m}{\phi }_s^{A,m}\left(y\right)\overline{{\mathfrak{F}}_A^m\left(\psi \right)\left(y\right)}d{\nu }_{A,b}\left(y\right)}\right)d{\gamma }_A\left(s\right).}$$

(31)

Therefore, by (7), Equation (31) is equal to

$${\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}\left({\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^2\overline{\psi \left(\xi \right)}d{\gamma }_A\left(\xi \right)}\right)d{\gamma }_A\left(s\right).}$$

Then

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\left({\displaystyle \frac{1}{{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}{\int }_{C(ϵ,\delta )}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\phi }_s^{A,m}\right)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(s\right)}\right)\overline{\psi \left(\xi \right)}d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\overline{\psi \left(\xi \right)}d{\gamma }_A\left(\xi \right).}\hfill \end{array}$$

(32)

Since by (7),

$${\displaystyle {\int }_{{\mathbb{R}}_{+}}{u}^{ϵ,\delta }\left(y\right)\overline{{\mathfrak{F}}_A^m\left(\psi \right)\left(y\right)}d{\nu }_{A,b}\left(y\right)={\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u^{ϵ,\delta }\right)\left(\xi \right)\overline{\psi \left(\xi \right)}d{\gamma }_A\left(\xi \right),}$$

(33)

then by (32) and (33),

$${\displaystyle {\int }_{{\mathbb{R}}_{+}}\left({\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u^{ϵ,\delta }\right)\left(\xi \right)-{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\right)\overline{\psi \left(\xi \right)}d{\gamma }_A\left(\xi \right)=0.}$$

Hence ${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u^{ϵ,\delta }\right)\left(\xi \right)={\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){\mathfrak{K}}_{ϵ,\delta }\left(\xi \right).$    □

Proof of Theorem 2.

From Lemma 3 and (9),

$$\begin{array}{ccc}\hfill \parallel u^{ϵ,\delta }{-u\parallel }_{L_{{\nu }_{A,b}}^2}^2& =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}(u^{ϵ,\delta }-u)\left(\xi \right)\right|}^{2}d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)({\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)-1)\right|}^{2}d{\gamma }_A\left(\xi \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){|}^2|1-{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\right|}^{2}d{\gamma }_A\left(\xi \right).}\hfill \end{array}$$

Moreover, from Lemma 2, we have for almost all $\xi \in {\mathbb{R}}_{+}$

$$\underset{ϵ\to 0,\delta \to \infty }{lim}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right|}^2{\left|1-{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\right|}^2=0,$$

and

$${\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right|}^2{\left|1-{\mathfrak{K}}_{ϵ,\delta }\left(\xi \right)\right|}^2\le C{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right)\right|}^2,$$

with $|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\left(\xi \right){|}^2\in L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)$. Thus, by the dominated convergence theorem, we get (27).    □

4. Practical Real Inversion Formulas

The theory of reproducing kernels has found wide applications in inverse problems, integral transforms, integral equations, inversion of bounded linear operators, sampling theory, differential equations with variable coefficients, and function approximation. Notable contributions in this area have been made by Saitoh et al. [30,32,33,34,35].

Before addressing applications to Tikhonov regularization, it is useful to examine Moore–Penrose generalized inverses through the framework of RKHS. This approach provides a natural and powerful tool for solving best approximation problems in Hilbert spaces.

Let E be a set and $H_K$ an RKHS on E with kernel K. Given a linear bounded operator $L:H_K\to \mathcal{H}$, where H is a Hilbert space, the classical best mean-square approximation problem is to find, for a given $d\in H$,

$$\underset{f\in {\mathcal{H}}_K}{inf}{|Lf-d|}_H.$$

In infinite-dimensional spaces, this problem is non-trivial and naturally leads to the concept of the Moore–Penrose generalized inverse. Its study involves both the existence of extremal functions and their explicit representation.

Tikhonov regularization introduces a positive parameter $r>0$ to ensure stability and uniqueness. Following Saitoh [30,32], one defines a modified inner product on $H_K$ by

$${\langle f_1,f_2\rangle }_{H_{K_r}}=r{\langle f_1,f_2\rangle }_{H_K^r}+{\langle L{f}_1,L{f}_2\rangle }_{\mathcal{H}},f_1,f_2\in H_K.$$

The resulting space $H_{K_r}$ is itself a Hilbert space with reproducing kernel $K_r$ given by

$$K_r(·,q)=(rI+L^{\ast }L)K(·,q),$$

where $L^{\ast }$ is the adjoint of $L:H_K\to \mathcal{H}$.

For every $h\in H$, the Tikhonov regularized extremal problem

$$\underset{f\in H_K}{inf}\left\{{r\parallel f\parallel }_{H_K^r}^2+{\parallel Lf-h\parallel }_{\mathcal{H}}^2\right\}$$

admits a unique solution $f_{r,h}^{\ast }\in H_K$, called the extremal function. This function has the explicit representation

$$f_{r,h}^{\ast }\left(p\right)={〈h,L{K}_r(·,p)〉}_{\mathcal{H}},p\in E.$$

This formulation demonstrates how reproducing kernel theory systematically yields Moore–Penrose generalized inverses and provides a constructive approach to best approximation problems in infinite-dimensional Hilbert spaces. It forms the theoretical foundation for applications such as Tikhonov regularization in the context of generalized Gabor transforms and other integral transforms.

In this paragraph, we apply the theory of reproducing kernels, with particular emphasis on Tikhonov regularization, to construct practical approximate solutions for equations involving bounded linear operators associated with the GGT.

4.1. Reproducing Kernels

For $s>0$, we introduce the generalized Paley–Wiener space by

$$W_{A,m}^s\left({\mathbb{R}}_{+}\right)=\left\{u\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right):\mathrm{supp}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(u\right)\subset [0,1/s]\right\}.$$

(34)

This type of space was first introduced by Paley and Wiener [36], and then studied extensively by Slepian and Pollak [37] in signal processing.

Lemma 4.

1. 

The generalized Paley–Wiener equipped with the following map ${\langle .,.\rangle }_{W_{A,m}^s}$ defined as

$${\displaystyle {\langle f,g\rangle }_{W_{A,m}^s}={\int }_0^{\frac{1}{s}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(g\right)\left(\lambda \right)}d{\gamma }_A\left(\lambda \right),f,g\in W_{A,m}^s\left({\mathbb{R}}_{+}\right),}$$

(35)

is an inner product space.

2. 

The generalized Paley–Wiener equipped with the following map ${\parallel ·\parallel }_{W_{A,m}^s}$ defined as

$${\parallel f\parallel }_{W_{A,m}^s}:={\left({\displaystyle {\int }_0^{\frac{1}{s}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)\right|}^{2}d{\gamma }_A\left(\lambda \right)}\right)}^{{\displaystyle \frac{1}{2}}},$$

is a normed space.

Proof. 

It is clear that $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ is a vector space. Moreover, in both statements, the difficulty is to demonstrate that we are dealing with a defined form, since the other axioms of product scalar and norm are satisfied. So, it suffices to prove that if ${\langle f,f\rangle }_{W_{A,m}^s}={\parallel f\parallel }_{W_{A,m}^s}=0$, then $f=0$. Indeed, if ${\parallel f\parallel }_{W_{A,m}^s}=0$, then

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)=0,\lambda \in [0,1/s].$$

(36)

As $f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$, then

$$\mathrm{supp}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\subset [0,1/s].$$

(37)

Involving (36) and (37), we derive that

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)=0,\mathrm{for}\mathrm{all}\lambda \in {\mathbb{R}}_{+}.$$

Using the fact that ${\mathfrak{F}}_A^m$ is an isometric isomorphism from $L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$ onto $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and (9), we derive that $f=0$.    □

Proposition 5.

$W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ admits the following reproducing kernel

$${\displaystyle {\mathfrak{I}}_{A,s}^m(x,y)={\int }_0^{\frac{1}{s}}{K}_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}d{\gamma }_{A,m}\left(\lambda \right),}$$

such that

1. 

For every $y\in {\mathbb{R}}_{+}$, ${\mathfrak{I}}_{A,s}^m(·,y)\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$;

2. 

For every $y\in {\mathbb{R}}_{+}$ and $f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$

$$f\left(y\right)={〈f,{\mathfrak{I}}_{A,s}^m(x,y)〉}_{W_{A,m}^s}.$$

Proof. 

For $y\in {\mathbb{R}}_{+}$, ${\Xi }_{y,s}^{A,m}(·)=\overline{K_A^m(·,y)}{1}_{(0,1/s)}(·)$ belongs to $L_{{\gamma }_A}^1\left({\mathbb{R}}_{+}\right)\cap L_{{\gamma }_A}^2\left({\mathbb{R}}_{+}\right)$. Then

$${\mathfrak{I}}_{A,s}^m(x,y)={\mathfrak{F}}_A^m\left({\Xi }_{y,s}^{A,m}\right)\left(x\right),x\in {\mathbb{R}}_{+}.$$

Moreover, from (4), ${\mathfrak{I}}_{A,s}^m(·,y)\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, and

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{I}}_{A,s}^m(·,y)\right)\left(\lambda \right)=\overline{K_A^m(\lambda ,y)}{1}_{(0,1/s)}\left(\lambda \right).$$

(38)

Using (6), we get

$$\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{I}}_{A,s}^m(·,y)\right)\left(\lambda \right)\right|\le 1_{(0,1/s)}\left(\lambda \right)$$

and

$$\parallel {\mathfrak{I}}_{A,s}^m{(·,y)\parallel }_{W_{A,m}^s}\le C(A,s):={\left({\int }_0^{{\displaystyle \frac{1}{s}}}d{\gamma }_A\left(\lambda \right)\right)}^{{\displaystyle \frac{1}{2}}}<\infty .$$

(39)

This shows that the function ${\mathfrak{I}}_{A,s}^m(·,y)$ is in $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$, for every $y\in {\mathbb{R}}_{+}$.

On the other hand, by (35) and (38),

$$\begin{array}{ccc}\hfill {\langle f,{\mathfrak{I}}_{A,s}^m(·,y)\rangle }_{W_{A,m}^s}& =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)\overline{{\left({\mathfrak{F}}_A^m\right)}^{-1}\left({\mathfrak{I}}_{A,s}^m(·,y)\right)\left(\lambda \right)}d{\gamma }_A\left(\lambda \right)}\hfill \\ & =& {\displaystyle {\int }_{{\mathbb{R}}_{+}}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)K_A^m(\lambda ,y)1_{(0,\frac{1}{s})}\left(\lambda \right)d{\gamma }_A\left(\lambda \right).}\hfill \end{array}$$

Using the fact that $f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$, we obtain $f\left(y\right)={〈f,{\mathfrak{I}}_{A,s}^m(x,y)〉}_{W_{A,m}^s}.$    □

Corollary 2.

$W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ is embedded in $C\left({\mathbb{R}}_{+}\right)$.

4.2. Extremal Functions

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $t\in {\mathbb{R}}_{+}$, we introduce the partial GGT by

$${\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right):={\mathfrak{G}}_{\phi }^{A,m}\left(f\right)(·,t),f\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right).$$

Proposition 6.

If $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$, then ${\mathfrak{V}}_{\phi ,t}^{A,m}$ is a linear bounded operator from $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ into $L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, such that

$$\parallel {\mathfrak{V}}_{\phi ,t}^{A,m}{\left(f\right)\parallel }_{L_{{\nu }_{A,b}}^2}\le {\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^1}{\parallel f\parallel }_{W_{A,m}^s},f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right).$$

Proof. 

The proof follows from (12) and (15).    □

For $r>0$, $s>0$, $t\in {\mathbb{R}}_{+}$ and $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$, we introduce the inner product in $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ by

$${〈f,g〉}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}=r{〈f,g〉}_{W_{A,m}^s}+{〈{\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right),{\mathfrak{V}}_{\phi ,t}^{A,m}\left(g\right)〉}_{L_{{\nu }_{A,b}}^2},$$

and its associated norm by

$${\parallel f\parallel }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}^2:=r{\parallel f\parallel }_{W_{A,m}^s}^2+{∥{\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right)∥}_{L_{{\nu }_{A,b}}^2}^2.$$

Remark 2.

A straightforward computation gives that the norms ${\parallel ·\parallel }_{W_{A,m}^s}$ and ${\parallel ·\parallel }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}$ are equivalent.

Proposition 7.

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$, the generalized Paley–Wiener space

$$\left(W_{A,m}^s\left({\mathbb{R}}_{+}\right),{\langle ·,·\rangle }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}\right),$$

has a reproducing kernel ${\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}$ given by

$${\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)={\left(rI+{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{-1}{\mathfrak{I}}_{A,s}^m(·,y),$$

where ${\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }:L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)⟶W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ is the adjoint of ${\mathfrak{V}}_{\phi ,t}^{A,m}$ defined by

$${\langle {\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right),g\rangle }_{L_{{\nu }_{A,b}}^2}={\langle f,{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }g\rangle }_{W_{A,m}^s},f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right),g\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right).$$

Moreover, we have:

1. 

For every $y\in {\mathbb{R}}_{+}$,

$${∥{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)∥}_{W_{A,m}^s}\le {\displaystyle \frac{C(A,s)}{r}}.$$

2. 

For every $y\in {\mathbb{R}}_{+}$,

$${∥{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{L_{{\nu }_{A,b}}^2}\le {\displaystyle \frac{C(A,s)}{\sqrt{2r}}}.$$

3. 

For all $y\in {\mathbb{R}}_{+}$,

$${∥{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{W_{A,m}^s}\le C(A,s),$$

where $C(A,s)$ is given by (39).

Proof. 

By Remark 2, Corollary 2, and Proposition 6, we derive that $v\mapsto v\left(t\right)$, $t\in {\mathbb{R}}_{+}$ is a continuous linear functional on $\left(W_{A,m}^s\left({\mathbb{R}}_{+}\right),{\langle ·,·\rangle }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}\right)$. Then, from [30], $\left(W_{A,m}^s\left({\mathbb{R}}_{+}\right),{\langle ·,·\rangle }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}\right)$ admits a reproducing kernel denoted by ${\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}$ and we have

$$\begin{array}{ccc}\hfill f\left(y\right)& =& {〈f,{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)〉}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}\hfill \\ & =& r{〈f,{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)〉}_{W_{A,m}^s}+{〈{\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right),{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)〉}_{L_{{\nu }_{A,b}}^2}\hfill \\ & =& {〈f,\left(rI+{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\right){\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)〉}_{W_{A,m}^s}.\hfill \end{array}$$

Thus,

$$\left(rI+{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\right){\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)={\mathfrak{I}}_{A,s}^m(·,y).$$

(40)

Furthermore, the previous identity implies that

$$\begin{array}{ccc}& & r^2{∥{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)∥}_{W_{A,m}^s}^2+2r{∥{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{L_{{\nu }_{A,b}}^2}^2+{∥{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{W_{A,m}^s}^2\hfill \\ & =& {∥{\mathfrak{I}}_{A,s}^m(·,y)∥}_{W_{A,m}^s}^2.\hfill \end{array}$$

Hence, since

$${∥{\mathfrak{I}}_{A,s}^m(·,y)∥}_{W_{A,m}^s}\le C(A,s),$$

we obtain the desired result.    □

Remark 3.

Following the approach of Proposition 5, we show that

$${\displaystyle {\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(x,y)={\int }_0^{\frac{1}{s}}\frac{K_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}d{\gamma }_A\left(\lambda \right).}$$

We now present the main result of this subsection.

Theorem 3.

Let $t\in {\mathbb{R}}_{+}$, $r>0$ and $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$. Then the following statements hold:

1. 

For every $g\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, there exists a unique best approximate function $f_{r,t,g}^{\ast }$ in the sense that it minimizes

$$\underset{f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)}{inf}\left\{{r\parallel f\parallel }_{W_{A,m}^s}^2+{\parallel g-{\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right)\parallel }_{L_{{\nu }_{A,b}}^2}^2\right\}.$$

(41)

Moreover, this extremal function admits the representation

$$f_{r,t,g}^{\ast }\left(y\right)={〈g,{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)〉}_{L_{{\nu }_{A,b}}^2},y\in {\mathbb{R}}_{+}.$$

(42)

2. 

The extremal function $f_{r,t,g}^{\ast }$ satisfies the estimate

$$\left|f_{r,t,g}^{\ast }\left(y\right)\right|\le {\displaystyle \frac{C(A,s)}{\sqrt{2r}}}{\parallel g\parallel }_{L_{{\nu }_{A,b}}^2}.$$

Proof. 

The existence and uniqueness of $f_{r,t,g}^{\ast }$ satisfying (41) follow from [32]. Moreover, it admits the representation

$$f_{r,t,g}^{\ast }\left(y\right)={〈g,{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)〉}_{L_{{\nu }_{A,b}}^2},y\in {\mathbb{R}}_{+}.$$

On the other hand, from Proposition 7,

$$|f_{r,t,g}^{\ast }{\left(y\right)|\le \parallel g\parallel }_{L_{{\nu }_{A,b}}^2}{∥{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{L_{{\nu }_{A,b}}^2}\le {\displaystyle \frac{C(A,s)}{\sqrt{2r}}}{\parallel g\parallel }_{L_{{\nu }_{A,b}}^2},$$

as desired.    □

Corollary 3.

If $f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ and $g={\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right)$, then for all $y\in {\mathbb{R}}_{+}$,

1. 

${\displaystyle f\left(y\right)=\underset{r\to 0^{+}}{lim}{f}_{r,t,g}^{\ast }\left(y\right)}$;

2. 

$\left|f\left(y\right)-f_{r,t,g}^{\ast }\left(y\right)\right|\le C(A,s){\parallel f\parallel }_{W_{A,m}^s}$;

3. 

$\left|f_{r,t,g}^{\ast }\left(y\right)\right|\le C(A,s){\parallel f\parallel }_{W_{A,m}^s}$.

Proof. 

Let f be in $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$. Then for all $y\in {\mathbb{R}}_{+}$,

$$f_{r,t,g}^{\ast }\left(y\right)={〈f,{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)〉}_{W_{A,m}^s}.$$

(43)

Therefore by (40),

$${\displaystyle \underset{r\to 0^{+}}{lim}{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)={\mathfrak{I}}_{A,s}^m(·,y).}$$

Thus

$$\underset{r\to 0^{+}}{lim}{f}_{r,t,g}^{\ast }\left(y\right)={〈f,{\mathfrak{I}}_{A,s}^m(·,y)〉}_{W_{A,m}^s}=f\left(y\right).$$

On the other hand, by (40) and (43), we have

$$f_{r,t,g}^{\ast }\left(y\right)=f\left(y\right)-r{〈f,{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)〉}_{W_{A,m}^s}.$$

Then, from Proposition 7,

$$\left|f_{r,t,g}^{\ast }\left(y\right)-f\left(y\right)\right|\le {r\parallel f\parallel }_{W_{A,m}^s}{∥{\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)∥}_{W_{A,m}^s}\le C(A,s){\parallel f\parallel }_{W_{A,m}^s}.$$

Finally, from Proposition 7 and (43), we have

$$\left|f_{r,t,g}^{\ast }\left(y\right)\right|\le {\parallel f\parallel }_{W_{A,m}^s}{∥{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)∥}_{W_{A,m}^s}\le C(A,s){\parallel f\parallel }_{W_{A,m}^s}.$$

The proof is complete.    □

Remark 4.

If ${\mathfrak{V}}_{\phi ,t}^{A,m}$ is an isometry, i.e., ${\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }{\mathfrak{V}}_{\phi ,t}^{A,m}=I$, then

1. 

${\langle ·,·\rangle }_{{\mathfrak{V}}_{\phi ,t}^{A,m},r,W_{A,m}^s}=(r+1){\langle ·,·\rangle }_{W_{A,m}^s};$

2. 

${\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(x,y)={\displaystyle \frac{1}{r+1}}{\mathfrak{I}}_{A,s}^m(x,y)$;

3. 

$f_{r,t,g}^{\ast }\left(y\right)={\displaystyle \frac{1}{r+1}}{\left({\mathfrak{V}}_{\phi ,t}^{A,m}\right)}^{\ast }g\left(y\right)$, where $g\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$;

4. 

$f_{r,t,{\mathfrak{V}}_{\phi ,t}^{A,m}\left(u\right)}^{\ast }\left(y\right)={\displaystyle \frac{1}{r+1}}u\left(y\right)$, where $u\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)$.

Proposition 8.

Let $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)\cap L_{{\nu }_{A,b}}^1\left({\mathbb{R}}_{+}\right)$.

1. 

For every $g\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$, the extremal function $f_{r,t,g}^{\ast }$ is represented by

$${\displaystyle f_{r,t,g}^{\ast }\left(x\right)={\int }_{\mathbb{R}}g\left(y\right){\mathfrak{Q}}_{r,t,g}^{A,m}(x,y)d{\gamma }_{A,m}\left(y\right),}$$

where

$${\displaystyle {\mathfrak{Q}}_{r,t,g}^{A,m}(x,y)={\int }_0^{\frac{1}{s}}{\displaystyle \frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{K}_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}}d{\gamma }_A\left(\lambda \right).}$$

(44)

2. 

If $g={\mathfrak{V}}_{\phi ,t}^{A,m}\left(f\right)$, then

$$\underset{r\to 0^{+}}{lim}{∥f_{r,t,g}^{\ast }-f∥}_{W_{A,m}^s}=0,$$

and ${\left\{f_{r,t,g}^{\ast }\right\}}_{r>0}$ converges uniformly as $r\to 0^{+}$ to f.

3. 

Let $\delta >0$ and let $g,g_{\delta }\in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ such that $\parallel g-g_{\delta }{\parallel }_{L_{{\nu }_{A,b}}^2}\le \delta$. Then

$${∥f_{r,t,g}^{\ast }-f_{r,t,g_{\delta }}^{\ast }∥}_{W_{A,m}^s}\le {\displaystyle \frac{\delta }{2\sqrt{r}}}.$$

Proof. 

From Theorem 3 and Remark 3, the infimum given by (42) is attained by a unique function $f_{r,t,g}^{\ast }$, and it is represented by

$$f_{r,t,g}^{\ast }\left(y\right)={〈g,{\mathfrak{G}}_{\phi }^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)(·,t)〉}_{L_{{\nu }_{A,b}}^2},y\in {\mathbb{R}}_{+},$$

where ${\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}$ is given in Remark 3. Moreover

$${\displaystyle {\mathfrak{G}}_{\phi }^{A,m}\left(f\right)(x,t)={\int }_0^{\frac{1}{s}}\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)K_A^m(\lambda ,x)d{\gamma }_A\left(\lambda \right).}$$

Then

$$\begin{array}{ccc}\hfill {\mathfrak{G}}_{\phi }^{A,m}\left({\mathfrak{J}}_{{\mathfrak{V}}_{\phi ,t}^{A,m},r}(·,y)\right)(·,t)\left(x\right)& =& {\displaystyle {\int }_0^{\frac{1}{s}}\frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{K}_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}d{\gamma }_A\left(\lambda \right)}\hfill \\ & =& {\mathfrak{Q}}_{r,t,g}^{A,m}(x,y).\hfill \end{array}$$

This gives (44). On the other hand, by (44),

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f_{r,t,g}^{\ast }\right)\left(\lambda \right)={\displaystyle \frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(g\right)\left(\lambda \right)}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}}.$$

Then

$${\left({\mathfrak{F}}_A^m\right)}^{-1}(f_{r,t,g}^{\ast }-f)\left(\lambda \right)={\displaystyle \frac{-r{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}}.$$

Thus

$${\displaystyle \parallel f_{r,t,g}^{\ast }{-f\parallel }_{W_{A,m}^s\left({\mathbb{R}}_{+}\right)}^2={\int }_{{\mathbb{R}}_{+}}{h}_{r,t,s}\left(\lambda \right){\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f\right)\left(\lambda \right)\right|}^{2}d{\gamma }_A\left(\lambda \right),}$$

with

$$h_{r,t,s}\left(\lambda \right)={\displaystyle \frac{r^2}{{\left(r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)\right)}^2}}{1}_{(0,1/s)}\left(\lambda \right).$$

Since

$$\underset{r\to 0}{lim}{h}_{r,t,s}\left(\lambda \right)=0$$

and

$$|h_{r,t,s}\left(\lambda \right)|\le 1,$$

and then we obtain the second result. Finally, by (44), we have

$${\left({\mathfrak{F}}_A^m\right)}^{-1}\left(f_{r,t,g}^{\ast }\right)\left(\lambda \right)={\displaystyle \frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(g\right)\left(\lambda \right)}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}}.$$

Therefore

$${\left({\mathfrak{F}}_A^m\right)}^{-1}(f_{r,t,g}^{\ast }-f_{r,t,g_{\delta }}^{\ast })\left(\lambda \right)={\displaystyle \frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{\left({\mathfrak{F}}_A^m\right)}^{-1}(g-g_{\delta })\left(\lambda \right)}{r+{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}}.$$

By the relation ${(x+y)}^2\ge 4xy$, we get

$$|{\left({\mathfrak{F}}_A^m\right)}^{-1}(f_{r,t,g}^{\ast }-f_{r,t,g_{\delta }}^{\ast })\left(\lambda \right){|}^2\le {\displaystyle \frac{1}{4r}}{\left|{\left({\mathfrak{F}}_A^m\right)}^{-1}(g-g_{\delta })\left(\lambda \right)\right|}^2.$$

Thus, by (9) we derive

$${∥f_{r,t,g}^{\ast }-f_{r,t,g_{\delta }}^{\ast }∥}_{W_{A,m}^s}^2\le {\displaystyle \frac{1}{4r}}{∥{\left({\mathfrak{F}}_A^m\right)}^{-1}(g-g_{\delta })∥}_{L_{{\nu }_{A,b}}^2}^2={\displaystyle \frac{1}{4r}}{∥g-g_{\delta }∥}_{L_{{\nu }_{A,b}}^2}^2,$$

which gives the desired result.    □

4.3. The Extremal Function Associated to the GGT

For $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$ and $r>0$, we introduce the inner product for the space $W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ by

$${\langle f,g\rangle }_{{\mathfrak{G}}_{\phi }^{A,m},r,W_{A,m}^s}=r{〈f,g〉}_{W_{A,m}^s}+{〈{\mathfrak{G}}_{\phi }^{A,m}\left(f\right),{\mathfrak{G}}_{\phi }^{A,m}\left(g\right)〉}_{L_{A,b}^2},$$

and its associated norm by

$${\parallel f\parallel }_{{\mathfrak{G}}_{\phi }^{A,m},r,W_{A,m}^s}^2:={r\parallel f\parallel }_{W_{A,m}^s}^2+{\parallel {\mathfrak{G}}_{\phi }^{A,m}\left(f\right)\parallel }_{L_{{\mu }_{A,b}}^2}^2.$$

By (17), we have

$${\langle f,g\rangle }_{{\mathfrak{G}}_{\phi }^{A,m},r,W_{A,m}^s}=r{\langle f,g\rangle }_{W_{A,m}^s}+{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2{\langle f,g\rangle }_{L_{{\nu }_{A,b}}^2}.$$

Using arguments similar to those in Proposition 7, Theorem 3, and Corollary 3, we obtain the following proposition.

Proposition 9.

Then the generalized Paley–Wiener space $\left(W_{A,m}^s\left({\mathbb{R}}_{+}\right),{\langle ·,·\rangle }_{{\mathfrak{G}}_{\phi }^{A,m},r,W_{A,m}^s}\right)$ has the following reproducing kernel

$${\displaystyle {\mathfrak{K}}_{r,\phi }^{A,m}(x,y)={\int }_0^{\frac{1}{s}}\frac{K_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}}{r+{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}d{\gamma }_A\left(\lambda \right),}$$

such that

1. 

${\mathfrak{K}}_{r,\phi }^{A,m}(x,y)={\left(rI+{\left({\mathfrak{G}}_{\phi }^{A,m}\right)}^{\ast }{\mathfrak{G}}_{\phi }^{A,m}\right)}^{-1}{\mathfrak{I}}_{A,s}^m(·,y),$ where ${\left({\mathfrak{G}}_{\phi }^{A,m}\right)}^{\ast }:L_{{\mu }_{A,b}}^2\left({\mathbb{R}}_{+}^2\right)⟶W_{A,m}^s\left({\mathbb{R}}_{+}\right)$ is the adjoint of ${\mathfrak{G}}_{\phi }^{A,m}$ defined by

$${\langle {\mathfrak{G}}_{\phi }^{A,m}\left(f\right),g\rangle }_{L_{{\mu }_{A,b}}^2}={\langle f,{\left({\mathfrak{G}}_{\phi }^{A,m}\right)}^{\ast }g\rangle }_{W_{A,m}^s},f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right),g\in L_{{\mu }_{A,b}}^2\left({\mathbb{R}}_{+}^2\right).$$

2. 

For every $y\in {\mathbb{R}}_{+}$,

$${∥{\mathfrak{K}}_{r,\phi }^{A,m}(·,y)∥}_{W_{A,m}^s}\le {\displaystyle \frac{C(A,s)}{r}}.$$

3. 

For every $y\in {\mathbb{R}}_{+}$,

$${∥{\mathfrak{G}}_{\phi }^{A,m}{\mathfrak{K}}_{r,\phi }^{A,m}(·,y)∥}_{L_{{\mu }_{A,b}}^2}\le {\displaystyle \frac{C(A,s)}{\sqrt{2r}}}.$$

4. 

For every $y\in {\mathbb{R}}_{+}$,

$${∥{\left({\mathfrak{G}}_{\phi }^{A,m}\right)}^{\ast }{\mathfrak{G}}_{\phi }^{A,m}\left({\mathfrak{K}}_{r,\phi }^{A,m}(·,y)\right)∥}_{W_{A,m}^s}\le C(A,s),$$

where $C(A,s)$ is given by (39).

Theorem 4.

Let $\phi \in L_{{\nu }_{A,b}}^2\left({\mathbb{R}}_{+}\right)$. Then the following statements hold:

1. 

For $r>0$ and $g\in L_{{\mu }_{A,b}}^2\left({\mathbb{R}}_{+}^2\right)$, there exists a unique best approximate function $f_{r,g}^{\ast }$ in the sense that it minimizes

$$\underset{f\in W_{A,m}^s\left({\mathbb{R}}_{+}\right)}{inf}\left\{{r\parallel f\parallel }_{W_{A,m}^s}^2+{\parallel g-{\mathfrak{G}}_{\phi }^{A,m}\left(f\right)\parallel }_{L_{{\mu }_{A,b}}^2}^2\right\}.$$

Moreover, this extremal function admits the representation

$${\displaystyle f_{r,g}^{\ast }\left(x\right)={〈g,{\mathfrak{G}}_{\phi }^{A,m}\left({\mathfrak{K}}_{r,\phi }^{A,m}(·,x)\right)〉}_{L_{{\mu }_{A,b}}^2}={\int }_{{\mathbb{R}}^2}g(t,y){\mathfrak{S}}_{r,\phi }^{A,m}(t,x,y)d{\mu }_{A,b}(t,y),}$$

where

$${\displaystyle {\mathfrak{S}}_{r,\phi }^{A,m}(t,x,y)={\int }_0^{\frac{1}{s}}\frac{\sqrt{{\tau }_t^A\left(\right|{\left({\mathfrak{F}}_A^m\right)}^{-1}\left(\phi \right){|}^2)\left(\lambda \right)}{K}_A^m(\lambda ,x)\overline{K_A^m(\lambda ,y)}}{r+{\parallel \phi \parallel }_{L_{{\nu }_{A,b}}^2}^2}d{\gamma }_A\left(\lambda \right).}$$

2. 

For g and $g_{\delta }$, $\delta >0$ such that $\parallel g-g_{\delta }{\parallel }_{L_{{\mu }_{A,b}}^2}\le \delta$, we have

$${∥f_{r,g}^{\ast }-f_{r,g_{\delta }}^{\ast }∥}_{W_{A,m}^s}\le {\displaystyle \frac{\delta }{2\sqrt{r}}}.$$

3. 

For $g={\mathfrak{G}}_{\phi }^{A,m}\left(f\right)$, we have

$$\underset{r\to 0^{+}}{lim}{\parallel f_{r,g}^{\ast }-f\parallel }_{W_{A,m}^s}=0,$$

and ${\left\{f_{r,g}^{\ast }\right\}}_{r>0}$ converges uniformly as $r\to 0^{+}$ to f.

4. 

For $g={\mathfrak{G}}_{\phi }^{A,m}\left(f\right)$, we have ${\displaystyle f\left(y\right)=\underset{r\to 0^{+}}{lim}{f}_{r,g}^{\ast }\left(y\right)}$, for every $y\in {\mathbb{R}}_{+}$.

5. 

For $g={\mathfrak{G}}_{\phi }^{A,m}\left(f\right)$, we have for every $y\in {\mathbb{R}}_{+}$,

$$|f\left(y\right)-f_{r,g}^{\ast }{\left(y\right)|\le C(A,s)\parallel f\parallel }_{W_{A,m}^s}$$

6. 

For $g={\mathfrak{G}}_{\phi }^{A,m}\left(f\right)$, we have for every $y\in {\mathbb{R}}_{+}$,

$$|f_{r,g}^{\ast }{\left(y\right)|\le C(A,s)\parallel f\parallel }_{W_{A,m}^s}.$$

5. Conclusions and Perspectives

In this paper, we have investigated several theoretical aspects of the generalized Gabor transform associated with a class of Sturm–Liouville operators in the framework of the Chébli–Trimèche hypergroup. In particular, we established inversion and Calderón-type formulas for this transform and studied some of its fundamental properties.

Furthermore, by applying the general theory of reproducing kernel Hilbert spaces, we analyzed approximation problems related to the generalized Gabor transform. Using the approach developed by Saitoh and others, we derived explicit representations of the extremal functions corresponding to certain regularized minimization problems. In this context, the Moore–Penrose generalized inverse and the Tikhonov regularization method were employed to obtain stable approximate solutions of bounded linear operator equations involving the generalized Gabor transform.

These results demonstrate that the theory of reproducing kernels provides a powerful and natural framework for the analysis of generalized integral transforms and related approximation problems.

As perspectives for future work, several directions may be explored. For instance, one may investigate further properties of the generalized Gabor transform in different functional spaces, such as weighted Sobolev or modulation-type spaces. Another possible direction is the study of uncertainty principles, sampling theorems, or frame properties associated with this transform. Moreover, applications of the generalized Gabor transform to inverse problems and signal analysis in non-Euclidean settings could provide interesting developments.