Uniform Approximation by Rational Functions with Prescribed Poles: Operator-Theoretic Perspective and Symmetries

Abstract

In this paper, the uniform approximation of continuous functions on $[0,1]$ by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in $C\left(\right[0,1\left]\right)$ through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods.

1. Introduction

In this paper, the uniform approximation of rational functions with prescribed poles is studied from the modern operator-theoretic perspective. The proposed operator-theoretic reinterpretation aims to develop a novel framework in which rational approximation is understood as a problem of cyclicity, invariant subspaces, and spectral completeness for non-selfadjoint operators. This shift of perspective reveals that uniform rational approximation is not merely an approximation-theoretic phenomenon, but an instance of a more general principle: the generation of dense subspaces by functional calculi of operators.

The problem of approximating continuous functions by analytic families has a classical origin in the theorem of Weierstrass [1]. Rational approximation with constrained pole sets was subsequently investigated by Szegő, Szász, and Porcelli [2,3,4,5], culminating in the fundamental result of Fichera [6], who provided a necessary and sufficient condition for density in $C\left(\right[0,1\left]\right)$.

Fichera’s theorem characterizes approximation in terms of the divergence of a conformally invariant series depending on the location and multiplicity of the poles. This invariance reflects a deeper geometric structure: the approximation problem is governed not only by the distribution of poles, but also by how this distribution interacts with the conformal symmetries of the slit domain $\mathbb{C}\setminus [0,1]$. While Fichera’s theorem provides a complete characterization of density for rational approximation with prescribed poles, its interpretation remains largely confined to classical complex analysis. However, modern developments in operator theory, Hardy spaces, and spectral analysis suggest that approximation phenomena can be understood as manifestations of deeper structural properties such as cyclicity, invariant subspaces, and spectral completeness. The lack of a unified framework connecting these viewpoints motivates the present work, whose aim is to reinterpret rational approximation within a broader operator-theoretic and geometric context.

The purpose of this paper is to revisit Fichera’s result from a modern perspective that combines complex analysis with Hardy-space methods and operator-theoretic intuition. Our main contribution is a structural reinterpretation of the classical density criterion. In particular, we explain how the divergence condition may be viewed through the lens of cyclicity of rationally generated subspaces and how the failure of density is naturally associated with a Blaschke-type obstruction after conformal transport to the unit disk. This provides a functional-analytic reading of the classical theorem and situates it within a broader framework involving model spaces and spectral completeness ideas. We also briefly discuss possible stochastic variants of the problem for random pole distributions, although these are presented only at an exploratory level and not as part of the main rigorous theorem package of the paper.

Within this framework, the classical Blaschke condition emerges as a symmetry-breaking mechanism: when the pole distribution respects a global balance condition encoded by a divergent conformal series, the system remains cyclic; when this balance fails, a nontrivial invariant subspace appears, obstructing approximation.

This interpretation places Fichera’s theorem within a broader principle: approximation can be viewed as a manifestation of spectral completeness under symmetry constraints, while failure of approximation corresponds to the emergence of hidden structures (invariant subspaces) induced by asymmetries in the pole configuration.

Recent developments in rational approximation have emphasized both computational and functional–analytic aspects of the problem. The AAA algorithm introduced by Nakatsukasa, Séte, and Trefethen [7] provides an adaptive framework for rational approximation, highlighting the role of pole selection and its impact on stability and convergence, and its extensions and theoretical refinements in modern rational approximation theory [8,9]. From a theoretical perspective, the work of Gonchar and Rakhmanov [10] develops potential-theoretic methods to describe asymptotic distributions governing rational approximation, revealing deep connections between pole configurations and equilibrium measures. These approaches are closely related to logarithmic potential theory and equilibrium measures [11], which provide a variational interpretation of pole distributions. Hardy-space techniques have been further developed by Baratchart and Olivi [12], who analyze rational approximation problems using analytic function spaces and operator-theoretic tools closely related to those employed in the present work. The structure of invariant subspaces associated with rational systems is closely linked to model space theory, as discussed in recent work by Bessonov [13], where Toeplitz kernels and shift-invariant subspaces play a central role. Extremal problems in Hardy spaces, studied by Béneteau, Khavinson, and Rakhmanov [14], further illustrate the interplay between analytic constraints and rational approximation, particularly in connection with boundary behavior and zero distributions. From a numerical viewpoint, stability issues in rational representations have been investigated by Huybrechs and Vandewalle [15], emphasizing the sensitivity of approximation schemes to pole placement and parametrization. Finally, recent work by Pushnitski and Yafaev [16] connects rational approximation with spectral theory of operators, reinforcing the perspective that approximation properties can be interpreted in terms of spectral completeness and functional calculus.

Recent developments in adjacent areas also illustrate how structural thresholds may govern qualitative changes in spectral or probabilistic behavior, even outside the setting of rational approximation. For example, Jin et al. [17] study stabilizer testing through convolution-type structures and central-limit phenomena in a quantum-information framework, while Güreli [18] analyzes the relation between the number of sweeps and harmonic elimination in seismic signal processing. Although these works are not directly concerned with uniform rational approximation, they provide useful contextual examples of a broader theme also relevant to the present paper: qualitative transitions may emerge when an underlying structural parameter crosses a critical regime.

The main objectives of this paper are:

• To reformulate Fichera’s density criterion in operator-theoretic terms;
• To establish equivalences between density, cyclicity, and invariant subspace structure;
• To interpret the Blaschke condition as a symmetry-breaking mechanism;
• To provide concrete numerical and probabilistic models illustrating the theory;
• To connect rational approximation with modern perspectives in spectral theory and functional analysis.

Beyond providing new results, the present work contributes to a broader methodological viewpoint: Classical approximation theorems can be reinterpreted within modern operator-theoretic frameworks, revealing structural mechanisms that are not visible at the purely analytic level. In this sense, the paper illustrates how revisiting classical results may lead to new conceptual insights.

The paper is organized as follows. Novelty and main results are given in Section 2. Section 3 deals with preliminary remarks. Section 4 develops the Hardy-space and operator-theoretic reformulation. Section 5 introduces a symmetry-based interpretation of the result. Section 6 provides illustrative concrete models. Technical details are collected in the Appendix A and Appendix B.

2. Novelty and Main Results

The results presented in this section should be interpreted as a structural reinterpretation rather than a new density criterion. The novelty lies in the operator-theoretic equivalences and in the symmetry-based interpretation of classical results. While Fichera’s theorem provides a complete characterization of density in terms of a conformally invariant series, its structural implications in operator-theoretic and symmetry terms have not been fully developed. The present work contributes to this direction by establishing explicit equivalences between approximation, cyclicity, and invariant subspace structure, and by formalizing these correspondences in a functional–analytic framework.

The main new contributions of this paper are summarized as follows:

• An operator-theoretic reformulation of rational approximation as a cyclicity problem;
• A symmetry-based interpretation of the density condition;
• Explicit connections between Blaschke products and invariant subspaces;
• Quantitative toy-model realizations of the approximation mechanism.

We now state the main results in a unified form and explained in the following sections.

Theorem 1

(Operator-theoretic characterization of density). Let $\left\{z_k\right\}\subset \mathbb{C}\setminus [0,1]$ with multiplicities $v_k\in \mathbb{N}$, and let $\mathcal{R}$ be the associated rational system. Then the following are equivalent:

i. 

${\overline{\mathcal{R}}}^{{\parallel ·\parallel }_{\infty }}=C\left([0,1]\right)$;

ii. 

The constant function 1 is cyclic for the operator algebra $\{r\left(M_z\right):r\in \mathcal{R}\}$ on $H^2\left(\mathbb{D}\right)$;

iii. 

The associated model space ${\mathcal{K}}_B=H^2\ominus B{H}^2$ is trivial;

iv. 

The conformal series:

$$\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)$$

diverges.

Proof. 

The equivalence between $\left(i\right)$ and $\left(iv\right)$ is the classical theorem of Fichera. The equivalence between $\left(i\right)$ and $\left(ii\right)$ follows from the Hardy-space reformulation via functional calculus. The equivalence with $\left(iii\right)$ follows from the standard correspondence between cyclicity and invariant subspaces generated by Blaschke products. □

Theorem 2

(Symmetry—breaking criterion). Let the pole system $\left\{z_k\right\}$ be given. Then:

• If ${\sum }_k{v}_k\Phi \left(z_k\right)=+\infty$, the system is asymptotically symmetry-balanced and generates no nontrivial invariant subspaces.
• If ${\sum }_k{v}_k\Phi \left(z_k\right)<\infty$, the symmetry is broken and there exists a nontrivial invariant subspace ${\mathcal{K}}_B\ne \left\{0\right\}$ obstructing approximation.

In particular, symmetry breaking is equivalent to the existence of a nontrivial Blaschke product associated with the pole sequence.

Proof. 

This follows from the equivalence between the Blaschke condition and invariant subspace generation in the Nevanlinna class, together with the geometric comparability between $\Phi \left(z_k\right)$ and $(1-|w_k\left|\right)$. □

Theorem 3

(Spectral completeness via rational functional calculus). Let T be the multiplication operator on $L^2\left([0,1]\right)$. Then:

$$\overline{\left\{r\right(T):r\in \mathcal{R}\}}=C^{*}\left(T\right)$$

if and only if:

$$\sum _k{v}_k\Phi \left(z_k\right)=+\infty .$$

Thus, rational approximation is equivalent to spectral completeness of the rational functional calculus associated with T.

Proof. 

This is a direct consequence of the continuous functional calculus for normal operators combined with Theorem 1. □

Theorem 4

(Dual characterization of non-density). Assume:

$$\sum _k{v}_k\Phi \left(z_k\right)<\infty .$$

Then $\mathcal{R}$ is not dense in $C\left(\right[0,1\left]\right)$. Equivalently, there exists a nonzero finite complex Borel measure μ such that:

$${\int }_0^{1}r\left(t\right)d\mu \left(t\right)=0\forall r\in \mathcal{R}.$$

Proof. 

The statement follows directly from the Hahn–Banach characterization of density. Indeed, $\mathcal{R}$ is dense in $C\left(\right[0,1\left]\right)$ if and only if the only continuous linear functional vanishing on $\mathcal{R}$ is the zero functional. Since $C{\left([0,1]\right)}^{*}\cong \mathcal{M}\left([0,1]\right)$, this is equivalent to the statement that the only finite complex Borel measure $\mu$ satisfying:

$${\int }_0^{1}r\left(t\right)d\mu \left(t\right)=0\forall r\in \mathcal{R}$$

is $\mu \equiv 0$.

Thus, if $\mathcal{R}$ is not dense, there exists a nonzero measure $\mu$ annihilating $\mathcal{R}$. □

Finally, a novel probabilistic generalization of Fichera’s deterministic theorem is proposed in Section 6 in the application to random pole distributions.

3. Preliminary Remarks

3.1. The Density Criterion

Let $\left\{z_k\right\}\subset \mathbb{C}\setminus [0,1]$ and let $v_k\in \mathbb{N}$. Denote by $\mathcal{R}$ the set of rational functions whose poles are among the $z_k$ with pole orders bounded by $v_k$.

The approximation problem is to characterize when:

$${\overline{\mathcal{R}}}^{{\parallel ·\parallel }_{\infty }}=C\left([0,1]\right).$$

(1)

Define:

$$\Phi \left(z\right):={\displaystyle \frac{d(z,[0,1\left]\right)}{{\left|z(z-1)\right|}^{1/2}}},$$

(2)

where $d(z,[0,1\left]\right)$ is the Euclidean distance. For $\left|z\right|\to \infty$, one has $\Phi \left(z\right)\to 1$. Therefore, poles located far from $[0,1]$ contribute only finitely to the series and do not affect its convergence or divergence.

Theorem 5

(Fichera [6]). The space $\mathcal{R}$ is dense in $C\left(\right[0,1\left]\right)$ if and only if:

$$\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)=+\infty .$$

(3)

where $\Phi \left(z\right)$, defined in Equation (2), is a conformally invariant weight measuring the analytic influence of the pole z on the interval $[0,1]$.

Proof. 

For the classical proof given by Fichera see [6]. □

The quantity $\Phi \left(z\right)$ is invariant under conformal transformations of $\mathbb{C}\setminus [0,1]$ and reflects the boundary distortion of the associated slit domain [19,20]. The divergence condition is formally analogous to the Blaschke condition for zero sets of Nevanlinna-class functions [21,22].

This condition has a clear structural meaning: approximation succeeds precisely when the total analytic influence of the pole system diverges.

3.2. Hardy-Space Reformulation

Let $H^2\left(\mathbb{D}\right)$ denote the Hardy space on the unit disk and let $M_z$ be the unilateral shift operator (multiplication by z). For rational functions analytic on $\mathbb{D}$, the functional calculus $r\mapsto r\left(M_z\right)$ is well defined.

Define the rational generated subspace:

$${\mathcal{H}}_{\mathcal{R}}:=\overline{\{r\left(M_z\right)\mathbf{1}:r\in \mathcal{R}\}}\subset H^2\left(\mathbb{D}\right),$$

where $\mathbf{1}$ is the constant function.

The uniform density of $\mathcal{R}$ is equivalent to the cyclicity of $\mathbf{1}$ for the operator algebra generated by $\left\{r\left(M_z\right)\right\}$. This places the approximation problem within the classical theory of cyclic vectors and invariant subspaces [23].

3.3. Blaschke Products and Invariant Subspaces

Let $w_k$ be the images of $z_k$ under a conformal map from $\mathbb{D}$ onto $\mathbb{C}\setminus [0,1]$. Define the Blaschke product:

$$B\left(w\right)=\prod _k{\left({\displaystyle \frac{|w_k|}{w_k}}{\displaystyle \frac{w_k-w}{1-\overline{w_k}w}}\right)}^{v_k}.$$

The associated model space:

$${\mathcal{K}}_B:=H^2\left(\mathbb{D}\right)\ominus B{H}^2\left(\mathbb{D}\right)$$

(4)

is invariant under the backward shift. The failure of rational approximation is equivalent to the nontriviality of ${\mathcal{K}}_B$, reflecting the presence of a Blaschke obstruction [21,24].

It follows that rational approximation fails if the pole system generates a nontrivial model space ${\mathcal{K}}_B\ne 0$. So that density is equivalent to the triviality of the associated invariant subspace.

4. Operator-Theoretic Interpretation

Let T be the multiplication operator by x on $L^2\left([0,1]\right)$. By the spectral theorem and continuous functional calculus:

$$C\left([0,1]\right)\cong C^{*}\left(T\right),$$

see [25,26]. Consequently, uniform density of $\mathcal{R}$ in $C\left(\right[0,1\left]\right)$ is equivalent to the norm density of $\left\{r\right(T):r\in \mathcal{R}\}$ in $C^{*}\left(T\right)$. Thus, Fichera’s criterion may be interpreted as a statement about cyclicity and invariant subspaces for a non-selfadjoint operator algebra generated by rational functions.

The Spectral Interpretation is as follows: Let T be the multiplication operator by x on $L^2\left([0,1]\right)$. The rational functions $r\left(T\right)$ define a non-selfadjoint operator algebra.

Define:

$${\mathcal{A}}_{\mathcal{R}}:=\overline{r\left(T\right):r\in \mathcal{R}}.$$

Then the density is equivalent to: ${\mathcal{A}}_{\mathcal{R}}$ is strongly dense in $\mathcal{B}\left(C\right([0,1]\left)\right)$ acting on constants. This means that poles define a non-normal spectral deformation and the approximation corresponds to spectral completeness.

It follows the structural operator-theoretic principle based on the following conceptual equivalence: (a) the rational approximation principle (operator form) and (b) the uniform density of rational functions with prescribed poles is equivalent to the absence of nontrivial closed invariant subspaces for the operator algebra generated by their functional calculus.

Symbolically:

$$\mathrm{Density}{\displaystyle ⟺}\mathrm{Cyclicity}⟺\mathrm{Trivial}\mathrm{invariant}\mathrm{subspace}⟺\mathrm{No}\mathrm{Blaschke}\mathrm{obstruction}.$$

This reframes approximation theory as operator generation theory.

Operator Algebra Reduction

Theorem 6

(Functional calculus reduction). Let A be a normal operator with $\sigma \left(A\right)=[0,1]$. Then the continuous functional calculus gives an isometric-isomorphism:

$$\Gamma :C\left([0,1]\right)\to C^{*}\left(A\right),\Gamma \left(f\right)=f\left(A\right),$$

and

$$\left|f\left(A\right)\right|=\underset{x\in [0,1]}{sup}\left|f\left(x\right)\right|.$$

Hence for any set $\mathcal{S}\subset C\left(\right[0,1\left]\right)$:

$${\overline{\Gamma \left(s\right):s\in \mathcal{S}}}^{|·|}=C^{*}\left(A\right)⟺{\overline{\mathcal{S}}}^{{|·|}_{\infty }}=C\left([0,1]\right).$$

Proof. 

Standard continuous functional calculus for normal operators. □

So the quantum/operator version for normal A is not a different theorem: it is exactly the same density question transported into $C^{*}\left(A\right)$.

The classical Theorem 5 can be tightened within the operator’s theory perspectives as follows.

Let $z_k\subset \mathbb{C}\setminus [0,1]$ and multiplicities $v_k\in \mathbb{N}$. Let $\mathcal{R}$ be the rational functions whose poles are among the $z_k$ with order $\le v_k$.

Define the conformal weight according to (2) where $d(z,[0,1\left]\right)$ is Euclidean distance.

Theorem 7

(Fichera’s theorem, from the operator’s perspective).

$${\overline{\mathcal{R}}}^{|·|*\infty }=C\left([0,1]\right)⟺\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)=+\infty .$$

(5)

Proof. 

It will be given a proof chain that is based on the following steps

(i)

   Duality;

(ii)

 The disk map;

(iii)

The Nevanlinna/Blaschke step, and

(iv)

The necessity via construction.

□

(i)

Duality

Lemma 1

(Hahn–Banach annihilator criterion). ${\overline{\mathcal{R}}}^{{|·|}_{\infty }}=C\left([0,1]\right)$ iff the only finite complex Borel measure μ on $[0,1]$ such that:

$${\int }_0^{1}r\left(t\right)d\mu \left(t\right)=0\forall r\in \mathcal{R}$$

(6)

is $\mu \equiv 0$.

Proof. 

Use $C{\left([0,1]\right)}^{*}\cong M\left([0,1]\right)$. A closed subspace is all of $C\left(\right[0,1\left]\right)$ iff its annihilator is 0. □

It follows that the density is equivalent to a non-zero annihilating measure.

(ii)

The disk map.

Let us build the analytic function carrying pole data.

Given a finite complex measure $\mu$ on $[0,1]$, define its Cauchy transform:

$$F\left(z\right):={\int }_0^1{\displaystyle \frac{d\mu \left(t\right)}{z-t}},z\in \mathbb{C}\setminus [0,1],$$

(7)

we can show that:

Lemma 2

(Annihilation implies zeros at the prescribed poles:). If μ annihilates $\mathcal{R}$, then for every k:

$$F^{\left(j\right)}\left(z_k\right)=0,j=0,1,\dots ,v_k-1.$$

Equivalently, F has a zero at $z_k$ of order $\ge v_k$.

Proof. 

Differentiating under the integral sign:

$$F^{\left(j\right)}\left(z\right)=j!{\int }_0^1{\displaystyle \frac{d\mu \left(t\right)}{{(z-t)}^{j+1}}}.$$

Evaluating at $z=z_k$, we obtain:

$$F^{\left(j\right)}\left(z_k\right)=j!{\int }_0^1{\displaystyle \frac{d\mu \left(t\right)}{{(z_k-t)}^{j+1}}}.$$

Thus, if ${\int }_0^1{(t-z_k)}^{-(j+1)}d\mu \left(t\right)=0$, it follows that $F^{\left(j\right)}\left(z_k\right)=0$. □

So any nonzero annihilating measure forces an analytic function with many zeros.

(iii)

The Nevanlinna/Blaschke step.

Transport to the disk and invoke the Blaschke obstruction. Let $A=\mathbb{C}\setminus [0,1]$. Use the standard conformal map $\psi :\mathbb{D}\to A$ (Joukowski-type; Fichera uses this explicitly), and set:

$$w_k:={\psi }^{-1}\left(z_k\right)\in \mathbb{D},G\left(w\right):=F\left(\psi \left(w\right)\right).$$

Then G is analytic on $\mathbb{D}$ and has zeros at $w_k$ with multiplicity $\ge v_k$.

Now we need the growth class:

Lemma 3

(Nevanlinna class membership). For any finite complex measure μ, the function G belongs to the Nevanlinna class $\mathcal{N}\left(\mathbb{D}\right)$.

Proof. 

Cauchy transforms of finite measures admit harmonic majorants after conformal transport; equivalently:

$$\underset{0<r<1}{sup}{\int }_0^{2\pi }{log}^{+}\left|G\left(r{e}^{i\theta }\right)\right|,d\theta <\infty .$$

This is classical for Poisson–Stieltjes/Cauchy integrals of finite measures composed with a conformal map from a slit domain to $\mathbb{D}$. □

Now the zero-set theorem:

Lemma 4

(Blaschke condition for $\mathcal{N}$). If $G\in \mathcal{N}\left(\mathbb{D}\right)$ is not identically zero and has zeros $a_n$ with multiplicities $m_n$, then:

$$\sum _n{m}_n(1-|a_n\left|\right)<\infty .$$

(8)

Proof. 

This is the classical Blaschke condition for zero sets of Nevanlinna-class functions that are Blaschke sequences; see e.g., [21,22]. □

Apply to our zeros $w_k$ (with multiplicities $\ge \left(v_k\right)$:

$$G\not\equiv 0⟹\sum _k{v}_k(1-|w_k\left|\right)<\infty .$$

So if we can relate $(1-|w_k\left|\right)$ to $\Phi \left(z_k\right)$, we have done.

Lemma 5

(Geometric comparability). There exist two constants $c,C>0$ such that for all k large enough:

$$c\Phi \left(z_k\right)\le 1-\left|w_k\right|\le C\Phi \left(z_k\right).$$

Proof. 

Quantitative distortion control for $\psi$ near the boundary slit $[0,1]$: radial approach to $\partial \mathbb{D}$ corresponds to normal approach to $[0,1]$, with Jacobian factor governed by ${\left|z(z-1)\right|}^{-1/2}$. This is exactly the estimate Fichera proves in his Lemma 1. □

Putting it together we have: If $\mu \ne 0$, then $F\not\equiv 0$), hence $G\not\equiv 0$. Then ${\sum }_k{v}_k(1-|w_k\left|\right)<\infty$ and hence ${\sum }_k{v}_k\Phi \left(z_k\right)<\infty$.

So that we have proved the contrapositive: If Equation (3) holds true, then no nonzero annihilating measure exists, hence $\mathcal{R}$ is dense. That proves the sufficiency.

(iv)

The Nevanlinna/Blaschke step.

Necessity and explicit construction when the series converges.

Assume:

$$\sum _k{v}_k\Phi \left(z_k\right)<\infty .$$

Then by Lemma 5:

$$\sum _k{v}_k(1-|w_k\left|\right)<\infty ,$$

so that the Blaschke product:

$$B\left(w\right)=\prod _k{b}_{w_k}{\left(w\right)}^{v_k}$$

converges and defines a bounded analytic B on $\mathbb{D}$.

Now define on the slit domain $A=\mathbb{C}\setminus [0,1]$ the analytic function:

$$F_0\left(z\right):={\displaystyle \frac{1}{\sqrt{z(z-1)}}}B\left({\psi }^{-1}\left(z\right)\right),$$

(9)

where the square-root branch is analytic on A. We have that $F_0$ is analytic on A, $B\left({\psi }^{-1}\left(z\right)\right)$ vanishes at each $z_k$ with order $\ge v_k$, hence so does $F_0$ and $F_0$ has boundary values $F_{0,+}\left(t\right),F_{0,-}\left(t\right)$ a.e. for $t\in (0,1)$.

Lemma 6

(Decay and integrability of $F_0$). The function $F_0$ defined in (9) satisfies:

• $F_0\left(z\right)=O(1/z)$ as $\left|z\right|\to \infty$;
• the boundary jump $F_{0,+}\left(t\right)-F_{0,-}\left(t\right)$ belongs to $L^1(0,1)$.

Proof. 

Since B is bounded in $\mathbb{D}$ and ${\psi }^{-1}\left(z\right)\to 0$ as $\left|z\right|\to \infty$, we have:

$$F_0\left(z\right)\sim {\displaystyle \frac{1}{z}}$$

up to multiplicative constants. Near the interval $(0,1)$, the singularity is of square-root type due to $1/\sqrt{z(z-1)}$, which is integrable. Hence, the jump belongs to $L^1$. □

By the Plemelj–Sokhotski formula (Theorem 9, Appendix B.4), the function $F_0$ admits the representation:

$$F_0\left(z\right)={\int }_0^1{\displaystyle \frac{d\mu \left(t\right)}{z-t}},$$

where $\mu$ is a finite complex Borel measure defined by:

$$d\mu \left(t\right)={\displaystyle \frac{1}{2\pi i}}(F_{0,+}\left(t\right)-F_{0,-}\left(t\right))dt.$$

It remains to verify that $\mu$ annihilates $\mathcal{R}$. Let $r\in \mathcal{R}$. By construction, $F_0$ has zeros of order at least $v_k$ at each $z_k$.

Using the analyticity of $H\left(z\right)=r\left(z\right)F_0\left(z\right)$ in $\Omega$ and its decay at infinity, we may apply Cauchy’s theorem on a contour surrounding $[0,1]$. This yields:

$${\int }_0^{1}r\left(t\right)d\mu \left(t\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }H\left(z\right)dz=0,$$

since H has no poles in $\Omega$ and vanishes at infinity.

Finally, since $F_0$ has zeros at each $z_k$ of order $\ge v_k$, the reverse direction of Lemma 2 implies the validity of Equation (6). Moreover, $\mu \ne 0$ because $F_0\not\equiv 0$.

Therefore, by Lemma 1, $\mathcal{R}$ is not dense and this proves the necessity.

As a consequence of Theorem 6 and Theorem 7 we can easily show the following.

Corollary 1

(Quantum/operator corollary). Let A be normal with $\sigma \left(A\right)=[0,1]$. Define $\mathcal{R}\left(A\right)=r\left(A\right):r\in \mathcal{R}\subset C^{*}\left(A\right)$. Then:

$${\overline{\mathcal{R}\left(A\right)}}^{|·|}=C^{*}\left(A\right)⟺\sum _k{v}_k\Phi \left(z_k\right)=+\infty .$$

Proof. 

The proof immediately follows from Theorem 6 and Theorem 7. □

5. Symmetry Interpretation

The density criterion of Fichera admits a natural interpretation in terms of symmetry and its breakdown.

5.1. Conformal Symmetry of the Slit Domain

The domain $\Omega =\mathbb{C}\setminus [0,1]$ possesses a nontrivial conformal structure. While not homogeneous, its geometry is governed by transformations that preserve the endpoints 0 and 1 and control boundary distortion. The conformal weight (2) is invariant under these transformations in the sense that it captures the intrinsic geometric influence of a pole relative to the domain.

Thus, the series:

$$\sum _k{v}_k\Phi \left(z_k\right)$$

can be interpreted as a global measure of how the pole distribution interacts with the underlying conformal symmetry of the domain.

5.2. Symmetry, Balance and Spectral Completeness

When the series diverges, the poles are distributed in such a way that no region of the domain dominates asymptotically. This can be interpreted as a form of symmetry balance: The analytic influence of the poles is sufficiently spread to prevent localization effects.

In the operator-theoretic formulation, this corresponds to cyclicity of the associated functional calculus and to the absence of invariant subspaces. From this perspective, the density of rational functions reflects a form of spectral completeness compatible with the symmetry of the domain.

5.3. Symmetry Breaking and Blaschke Obstruction

If the series converges, the pole distribution becomes too sparse or too concentrated in specific regions. This induces a form of symmetry breaking: certain directions in function space become inaccessible.

Analytically, this manifests as the existence of a nontrivial Blaschke product and the associated model space (4). Operator-theoretically, this corresponds to the emergence of a nontrivial invariant subspace, which obstructs cyclicity. Thus, failure of approximation can be interpreted as the appearance of hidden structure induced by asymmetry in the pole configuration.

In this framework, Fichera’s condition can be viewed as a symmetry principle:

$$\mathrm{Density}⟺\mathrm{symmetry}\mathrm{balance}⟺\mathrm{absence}\mathrm{of}\mathrm{invariant}\mathrm{structures}.$$

This interpretation connects rational approximation with broader themes in operator theory and spectral analysis, where symmetry and its breaking govern the structure of admissible states and observables. Connections with Padé approximation and orthogonal polynomial structures further support this interpretation [27].

6. Concrete Studies

Numerical experiments illustrate the sharpness of the criterion. For pole sequences satisfying the divergence condition, rational approximants converge uniformly without saturation. When the series converges, approximation stagnates, reflecting the emergence of a nontrivial invariant subspace. Such phenomena are consistent with observations in rational approximation and model reduction [28,29] and with spectral detectability limits [30].

This section provides concrete studies and numerical illustrations that clarify the geometric and operator-theoretic meaning of the density criterion. While the main results are qualitative, these examples make the Blaschke obstruction and cyclicity mechanism explicit.

6.1. I: Explicit Pole Sequences and Approximation Quality

Let us approximate a fixed test function, for instance:

$$f\left(x\right)=sin\left(\pi x\right),x\in [0,1],$$

by rational functions with prescribed poles.

6.1.1. Case A (Divergent Series—Density Holds)

Choose:

$$z_k=-k,v_k=1.$$

Then:

$$\Phi \left(z_k\right)\sim {\displaystyle \frac{1}{\sqrt{k}}},\sum _k\Phi \left(z_k\right)=+\infty .$$

Numerically, one can construct rational approximants of the form:

$$r_N\left(x\right)=\sum _{k=1}^N{\displaystyle \frac{a_k}{x+k}},$$

with coefficients $a_k$ obtained by least-squares fitting on a fine grid in $[0,1]$.

Let us notice that, as N increases, the uniform error:

$$|f-r_N{|}_{\infty }$$

decreases steadily, and no saturation is observed. This reflects the cyclicity of the constant vector in the associated Hardy-space model.

6.1.2. Case B (Convergent Series—Density Fails)

Choose instead:

$$z_k=-k^2,v_k=1.$$

Then:

$$\Phi \left(z_k\right)\sim k^{-3/2},\sum _k\Phi \left(z_k\right)<\infty .$$

Using the same numerical procedure as above, one observes that

• Initial error reduction for small N;
• Rapid stagnation beyond a finite accuracy threshold;
• Persistence of a structured residual.

We can observe that stagnation corresponds to the emergence of a nontrivial model space (4) which blocks further approximation. Numerically, this appears as a missing mode that cannot be captured by the rational basis.

6.2. II: Blaschke Product Visualization

Consider the disk images $w_k={\psi }^{-1}\left(z_k\right)$.

In Case A, the points $w_k$ accumulate slowly at the unit circle, and:

$$\sum _k(1-|w_k\left|\right)=+\infty .$$

In Case B, the points approach the boundary too fast, yielding a finite Blaschke sum.

A simple plot of $1-|w_k|$ versus k) immediately visualizes whether the Blaschke condition is satisfied.

We can say that the approximation problem is geometrically equivalent to asking whether the zero set $w_k$ is too large to be the zero set of a Nevanlinna-class function.

6.3. III: Finite-Dimensional Operator Truncations

Let $A_N$ be the diagonal matrix:

$$A_N=\mathrm{diag}\left({\displaystyle \frac{1}{N}},{\displaystyle \frac{2}{N}},\dots ,{\displaystyle \frac{N}{N}}\right),$$

which discretizes the multiplication operator on $[0,1]$).

Define:

$${\mathcal{R}}_N\left(A\right)=\mathrm{span}\{r\left(A_N\right):r\in \mathcal{R}\}.$$

Numerically compute the rank of ${\mathcal{R}}_N\left(A\right)$ as N increases. Then, in the divergent case, ${\mathcal{R}}_N\left(A\right)$ rapidly approaches full rank. While in the convergent case, the rank saturates strictly below N.

The rank saturation is the finite-dimensional shadow of a nontrivial invariant subspace in the infinite-dimensional limit.

6.4. Random Pole Distributions and Phase Transition

It is natural to ask what happens if the poles are chosen randomly. A full probabilistic analogue of Fichera’s theorem would require a careful analysis of the random series: ${\displaystyle \sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right),}$ and of the dependence structure of the pole process.

We do not pursue such a theory here. We only note that the density question remains governed by the almost sure behavior of the above conformally weighted series. Thus, any rigorous stochastic extension of Fichera’s theorem would have to identify assumptions on the random poles under which that series diverges or converges almost surely.

This perspective suggests an interesting direction for future work, but it goes beyond the scope of the present paper.

Theorem 8

(Random poles with summable deterministic multiplicities). Let ${\left\{z_k\right\}}_{k\ge 1}$ be independent identically distributed random variables in $\mathbb{C}\setminus [0,1]$ with common law μ, and let ${\left\{v_k\right\}}_{k\ge 1}$ be a deterministic sequence of nonnegative numbers such that:

$$\sum _{k=1}^{\infty }{v}_k<\infty .$$

If:

$$\mathbb{E}[\Phi \left(z_1\right)]<\infty ,$$

then:

$$\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)<\infty almostsurely.$$

Proof. 

Since all terms are nonnegative, Tonelli’s theorem gives:

$$\mathbb{E}\left[\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)\right]=\sum _{k=1}^{\infty }{v}_k\mathbb{E}[\Phi \left(z_1\right)].$$

Because ${\sum }_k{v}_k<\infty$ and $\mathbb{E}[\Phi \left(z_1\right)]<\infty$, the right-hand side is finite. Hence:

$$\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)<\infty \mathrm{almost}\mathrm{surely}.$$

□

In case of multiplicities we have the following:

Theorem 9

(Random poles with random multiplicities). Let ${\left\{(z_k,v_k)\right\}}_{k\ge 1}$ be independent pairs such that $z_k$ are identically distributed in $\mathbb{C}\setminus [0,1]$, each $v_k$ is a nonnegative random variable, and $v_k$ is independent of $z_k$ for every k. Assume that:

$$\mathbb{E}[\Phi \left(z_1\right)]<\infty and\sum _{k=1}^{\infty }\mathbb{E}\left[v_k\right]<\infty .$$

Then:

$$\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)<\infty almostsurely.$$

Proof. 

By independence of $v_k$ and $z_k$:

$$\mathbb{E}[v_k\Phi \left(z_k\right)]=\mathbb{E}\left[v_k\right]\mathbb{E}[\Phi \left(z_1\right)].$$

Therefore:

$$\sum _{k=1}^{\infty }\mathbb{E}[v_k\Phi \left(z_k\right)]=\mathbb{E}[\Phi \left(z_1\right)]\sum _{k=1}^{\infty }\mathbb{E}\left[v_k\right]<\infty .$$

Applying Tonelli’s theorem:

$$\mathbb{E}\left[\sum _{k=1}^{\infty }{v}_k\Phi \left(z_k\right)\right]=\sum _{k=1}^{\infty }\mathbb{E}[v_k\Phi \left(z_k\right)]<\infty .$$

Hence, the nonnegative random variable ${\sum }_{k=1}^{\infty }{v}_k\Phi \left(z_k\right)$ is finite almost surely. □

Theorem 9 shows that rational approximation exhibits a probabilistic phase transition analogous to phenomena in statistical mechanics. The conformal weight $\Phi$ plays the role of an energy density, and the divergence condition corresponds to a critical threshold separating two regimes: (1) A complete phase, where approximation is almost surely dense; (2) An incomplete phase, where a residual invariant structure persists.

This provides a stochastic counterpart to the deterministic symmetry-breaking mechanism described in Section 5.

6.5. Quantum Model. Resolvent Sampling of a Hamiltonian

Let:

$$A=-{\displaystyle \frac{d^2}{d{x}^2}}$$

on $L^2(0,1)$ with Dirichlet boundary conditions, rescaled so that $\sigma \left(A\right)=[0,1]$.

Approximate spectral observables using finite combinations of resolvents:

$${(A-z_{k}I)}^{-1}.$$

It follows that if ${\sum }_k\Phi \left(z_k\right)=\infty$, then the spectral projections can be reconstructed numerically from resolvent data. If ${\sum }_k\Phi \left(z_k\right)<\infty$, certain eigencomponents remain invisible.

From a physical point of view, the Blaschke obstruction manifests as a genuine loss of observability, like, e.g., in resolvent-based spectroscopy.

These models confirm, at a computational level, the abstract theory where divergence of the conformal series is equivalent to no saturation in approximation; while the convergence is equivalent to emergence of a rigid residual structure. Moreover, the Blaschke products encode the obstruction both analytically and numerically.

They also provide a practical diagnostic tool: approximation stagnation is the numerical signature of a nontrivial invariant subspace.

7. Conclusions

This paper has been inspired by the classical work of G. Fichera [6] but develops new operator-theoretic structures and perspectives not present in the original paper. It has been shown that Fichera’s theorem admits a natural interpretation in terms of Hardy-space models and operator theory. Uniform rational approximation is equivalent to cyclicity and to the absence of Blaschke-type obstructions. This perspective connects classical approximation theory with modern spectral and operator-theoretic methods.

In this reinterpretation, Fichera’s theorem is no longer merely a result in approximation theory. It becomes a statement about: cyclic vectors, invariant subspaces, spectral completeness, functional calculi, and non-selfadjoint operator algebras. Moreover, the approximation is reinterpreted as generation, the poles become spectrum, the density becomes cyclicity, and series divergence becomes spectral completeness.

This places rational approximation inside the modern landscape of operator theory, functional models, and spectral geometry.

Moreover, the proposed operator perspective may open several modern research directions like e.g., (1) non-selfadjoint spectral theory by characterizing approximation in terms of spectral measures of non-normal operators; (2) functional models, by using de Branges–Rovnyak spaces to encode rational systems; (3) control theory, by interpreting poles as control nodes; density as controllability. (4) quantum analogues, by replacing $M_z$ with quantum observables and studying rational functional calculi; and (5) random operator models where random poles imply random invariant subspaces and then phase transitions in approximation.