Toeplitz Localization Operators: Spectral Functions Density

Abstract

We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In this case, the Toeplitz localization operators (TLOs) exhibit an explicit diagonalization, i.e., there exists an isometric isomorphism that transforms all TLOs to the multiplication operators by some specific functions—we call them spectral functions. We show that these spectral functions can be written in the form of a convolution of the generating symbol of TLO with a kernel function incorporating an admissible wavelet/window. Using the Wiener’s deconvolution technique on the real line, we prove that the set of spectral functions is dense in the C$^*$-algebra of bounded uniformly continuous functions on the real line under the assumption that the Fourier transform of the kernel function does not vanish on the real line. This provides an explicit and independent description of the C$^*$-algebra generated by the set of spectral functions. The result is then applied to the case of a parametric family of wavelets related to Laguerre functions. Thereby we also provide an explicit description of the C$^*$-algebra generated by vertical Toeplitz operators on true poly-analytic Bergman spaces over the upper half-plane.

1 Introduction

Motivation According to [10], there are three non-trivial model classes of symbols such that the corresponding Toeplitz operators generate commutative algebras of operators on the Bergman spaces over the unit disc, or over the upper half-plane: elliptic (realized by radial symbols, i.e., functions depending only on |z| on the unit disc), parabolic (realized by vertical symbols, i.e., functions depending only on $\text {Im}(z)$ on the upper half-plane) and hyperbolic (realized by angular symbols, i.e., functions depending only on $\text {arg}\,z$ on the upper half-plane). In each one of these three cases there exists an isometric isomorphism that transforms all Toeplitz operators of the corresponding C$^*$-algebra to the multiplication operators by some specific functions. An isometric characterization of the elliptic, parabolic and hyperbolic case commutative algebra of Toeplitz operators reads as follows:

• (i) the C$^*$-algebra generated by the set of Toeplitz operators with bounded radial symbols is isometrically isomorphic to the C$^*$-algebra ${\text {VSO}}(\mathbb {N})$ of slowly oscillating sequences, see [2, 9, 31];

• (ii) the C$^*$-algebra generated by the set of Toeplitz operators with bounded vertical symbols is isometrically isomorphic to the C$^*$-algebra ${\text {VSO}}(\mathbb {R}_+)$ of very slowly oscillating functions on $\mathbb {R}_+$, see [11, 12];

• (iii) the C$^*$-algebra generated by the set of Toeplitz operators with bounded angular symbols is isometrically isomorphic to the C$^*$-algebra ${\text {VSO}}(\mathbb {R})$, see [5].

Recall that ${\text {VSO}}(\mathbb {R}_+)$ consists of all such bounded functions $\mathbb {R}_+\rightarrow \mathbb {C}$ that are uniformly continuous with respect to the metric $\rho (x,y)=|\ln x-\ln y|$ on $\mathbb {R}_+$. Note that the C$^*$-algebras ${\text {VSO}}(\mathbb {R}_+)$ and ${\text {VSO}}(\mathbb {R})$ corresponding to the vertical and angular cases, respectively, are isometrically isomorphic to the C$^*$-algebra $C_u(\mathbb {R})$ of bounded uniformly continuous functions on the real line. Moreover, the sequences from ${\text {VSO}}(\mathbb {N})$ are nothing but the restrictions of functions from ${\text {VSO}}(\mathbb {R}_+)$ to natural numbers $\mathbb {N} = \{1,2,\ldots \}$.

This paper focuses on studying the C$^*$-algebras generated by Toeplitz localization operators. Operators of that kind have been extensively studied by many authors, see, for example, [3, 4, 6, 19, 25, 27, 30]. We aim to give a time-frequency analogy of the above mentioned description of commutative C$^*$-algebras of Toeplitz operators on Bergman spaces. For that reason we shall briefly recall some necessary notions and notation.

Toeplitz localization operators A starting point for the construction of time-frequency localization (filter) operators are the reproducing formulas of Calderón in wavelet analysis and of Gabor in time-frequency analysis. They can be written in a unified form as follows (see Sect. 2 for more details): for every $f\in L_2(\mathbb {R})$,

$$\begin{aligned} f=\int _{\mathbb {X}} \langle f,\Psi _{\zeta }\rangle \Psi _{\zeta }\,\mathrm {d}\zeta , \end{aligned}$$

where $\mathbb {X}$ is either the affine group and $\Psi$ is an admissible wavelet (in wavelet case), or $\mathbb {X}$ is the whole plane and $\Psi$ is an admissible window (in time-frequency case). The transform $W_{\Psi }: L_2(\mathbb {R})\rightarrow L_{2}(\mathbb {X},\,\mathrm {d}\zeta )$ defined by $(W_{\Psi } f)(\zeta )=\langle f,\Psi _{\zeta }\rangle$ is known as the continuous wavelet transform (in wavelet case), or short-time Fourier transform (in time-frequency case). The image $W_{\Psi }(L_{2}(\mathbb {R}))$ is a closed subspace of the Hilbert space $L_{2}(\mathbb {X},\,\mathrm {d}\zeta )$. It is called the space of wavelet transforms, or short-time Fourier transforms.

For a given bounded function a on $\mathbb {X}$, the Toeplitz localization operator (TLO) $T_a^{\Psi }$ with defining symbol a acts on $W_{\Psi }(L_{2}(\mathbb {R}))$ by the rule $T_a^\Psi f =P_{\Psi }(af)$, where $P_{\Psi }$ stands for the orthogonal projection from $L_2(\mathbb {X}, \mathrm {d}\zeta )$ onto $W_{\Psi }(L_{2}(\mathbb {R}))$. In the wavelet case, the Toeplitz operator $T_a^\Psi$ is usually called the Calderón-Toeplitz operator, whereas in the time-frequency case the operator $T_a^\Psi$ is usually called the Gabor-Toeplitz operator. In this paper, we restrict our attention to TLOs with generating symbols a depending only on the first coordinate in the phase space, and we denote this set of operators by $\mathfrak {T}_\Psi$. As shown in [17], for each fixed admissible $\Psi$ there exists a unitary operator $R_{\Psi }$ that diagonalizes each TLO $T_a^\Psi$, i.e., $R_\Psi T_a^\Psi R_\Psi ^* = \gamma ^\Psi _{a} I$ with a certain explicitly given function $\gamma ^\Psi _a$ called the spectral function. This simultaneous diagonalization of the operators from the class $\mathfrak {T}_\Psi$ ensures that the C$^*$-algebra $\mathcal {T}_\Psi$ generated by $\mathfrak {T}_\Psi$ is commutative and provides direct access to their properties: norms, boundedness, spectra, invariant subspaces, etc. Thus, the operators from the set $\mathfrak {T}_\Psi$ can be considered as “domestic”, i.e., simple to study and understand. This knowledge extends naturally to other operators that can be diagonalized in the same way and approximated by $\mathfrak {T}_\Psi$ in the uniform operator topology, or even to the C$^*$-algebra $\mathcal {T}_\Psi$ generated by $\mathfrak {T}_\Psi$. Under certain additional hypotheses, the class of such domestic operators can be explicitly described by providing a simple description of the C$^*$-algebra $\mathcal {G}_\Psi$ generated by the corresponding spectral functions, as it is stated in what follows. Table 1 brings together the notation that we use for the operators, their spectral functions, and the corresponding sets and C$^*$-algebras.

Table 1 Passing from operators to their spectral functions

Main results We first show that each spectral function $\gamma ^\Psi _{a}$ corresponding to TLO $T_a^\Psi$ with a bounded generating symbol a on $\mathbb {R}$ is a convolution on the real line with an explicitly given kernel $K_\Psi$. Then using Wiener’s deconvolution on the real line we get the following result:

Theorem 1

If the Fourier transform $\widehat{K}_\Psi$ of a convolution kernel $K_\Psi \in L_1(\mathbb {R})$ does not vanish on $\mathbb {R}$, then the set

$$\begin{aligned} \mathfrak {G}_\Psi = \left\{ K_\Psi *a;\,\, a\in L_\infty (\mathbb {R})\right\} \end{aligned}$$

is a dense subset of $C_u(\mathbb {R})$. Consequently, the C$^*$-algebra $\mathcal {T}_\Psi$ generated by the set $\mathfrak {T}_\Psi$ is isometrically isomorphic to $C_u(\mathbb {R})$.

It follows that the uniform closure of the set $\mathfrak {T}_\Psi$ coincides with the C$^*$-algebra $\mathcal {T}_\Psi$ generated by the set $\mathfrak {T}_\Psi$. Furthermore, there is no qualitative difference between the wavelet/window cases when studying the structure of commutative algebra $\mathcal {T}_\Psi$. Indeed, under the assumption that $\widehat{K}_\Psi$ does not vanish on $\mathbb {R}$, each C$^*$-algebra $\mathcal {T}_\Psi$ generated by TLOs with symbols $a\in L_\infty (\mathbb {R})$ coincides with the closure of the set of its initial generators. Furthermore, in that case Theorem 1 gives an isometric characterization of the “time-frequency” case commutative algebra of TLOs as an analogy of the description of commutative C$^*$-algebras of Toeplitz operators with radial, vertical and angular symbols acting on Bergman spaces. The proof of Theorem 1 is more general, but not so explicit in comparison with the constructions used in the proofs of density in [11] and [12]. Note that the method used in the proof of Theorem 1 yields an elegant and short proof of the main result of [11].

There is another connection between [12] and Theorem 1. Surprisingly, it is described in [18] that the case of Toeplitz operators acting on true poly-analytic Bergman spaces can be viewed as TLOs related to a special case of wavelets, namely, to the wavelets constructed from Laguerre functions on the half-line. It is also known from [18] that the C$^*$-algebra generated by Toeplitz operators with bounded vertical symbols acting on true poly-analytic Bergman space of order $k\in \mathbb {Z}_+ = \mathbb {N}\cup \{0\}$ over the upper half-plane is commutative, and is isometrically isomorphic to the C$^*$-algebra generated by the set

$$\begin{aligned} \mathfrak {G}_k := \{\gamma _{a,k};\,a\in L_\infty (\mathbb {R}_+)\} \end{aligned}$$

where the corresponding spectral functions are given by

$$\begin{aligned} \gamma _{a,k}(u) := \int _{\mathbb {R}_+} a\left( \frac{v}{2u}\right) \mathrm {e}^{-v} L_k^2(v)\,\mathrm {d}v, \end{aligned}$$

with $L_k(x)$ being the Laguerre polynomial of order k. Thus, as a particular case of Theorem 1, we extend the characterization of the C$^*$-algebra generated by vertical Toeplitz operators acting on the non-weighted Bergman space of analytic functions from [12] to the true poly-analytic case proving the following density result:

Theorem 2

For each $k\in \mathbb {Z}_+$, the set $\mathfrak {G}_k$ is dense in the C$^*$-algebra ${\text {VSO}}(\mathbb {R}_+)$.

Theorem 2 with $k=0$ is the main result of [12]. As a consequence of Theorem 2 we have that the C$^*$-algebra $\mathcal {T}_k$ generated by Toeplitz operators acting on true poly-analytic Bergman space of order k with bounded vertical symbols is isometrically isomorphic to ${\text {VSO}}(\mathbb {R}_+)$. Moreover, Theorem 2 shows that the set of initial generators of $\mathcal {T}_k$ is dense in $\mathcal {T}_k$, i.e., the C$^*$-algebraic closure of the set of initial generators coincides with its topological closure. These results do not depend on the value of the weight parameter k.

Note that the C*-algebra generated by all Toeplitz operators with bounded symbols, acting on the Bergman space over the unit ball in $\mathbb {C}^n$, also coincides with the topological closure of the set of initial generators. This result was recently proved by Xia [34].

Organization of paper Section 2 summarizes the basic facts on time-frequency analysis tools used in the paper, Toeplitz localization operators and their diagonalization. In Sect. 3 we prove Theorem 1 using Wiener’s division lemma. The special case of Toeplitz operators related to wavelets constructed from Laguerre functions on the half-line is investigated in Sect. 4, where a detailed proof of Theorem 2 is given. The main concept of the proof of Theorem 2 is to move from the half-line to the real line and to use the approximate deconvolution technique developed in Sect. 3. Verification that the Fourier transform of the corresponding convolution kernel does not vanish on the real line (i.e., the assumption of Theorem 1) is a fine part of the proof and leads to investigation of zeros of a family of certain complex polynomials related to symmetric Meixner-Pollaczek polynomials. We solve this task using techniques from the theory of orthogonal polynomials.

2 Preliminaries

In this section we recall some basic facts we work with in this paper. As a standard overview on the rich subject of time-frequency analysis methods, concepts and basic results we recommend the books [8, 13, 33].

Reproducing formulas of time-frequency analysis in a unified setting In this paper we work with both the reproducing formulas of Calderón in wavelet analysis and of Gabor in time-frequency analysis, using the following notation: for every $f\in L_2(\mathbb {R})$,

$$\begin{aligned} f=\int _{\mathbb {X}} \langle f,\Psi _{\zeta }\rangle \Psi _{\zeta }\,\mathrm {d}\zeta . \end{aligned}$$

(1)

In the wavelet case, $\mathbb {X}$ is the affine group $\mathbb {G}=\{\zeta =(u,v); \,\,u>0,\ v\in \mathbb {R}\}$ with hyperbolic metric and corresponding hyperbolic measure $\mathrm {d}\zeta = u^{-2}\,\mathrm {d}u\,\mathrm {d}v$; and $\Psi =\psi$ is a wavelet, i.e., a function from $L_2(\mathbb {R})$ satisfying the admissibility condition

$$\begin{aligned} \int _{\mathbb {R}_{+}}|\widehat{\psi }(t\xi )|^{2}\,\frac{\mathrm {d}t}{t}=1\quad \text {for a.e.}\quad \xi \in \mathbb {R}, \end{aligned}$$

where $\widehat{f}$ is the Fourier transform $\mathcal {F}: L_{2}(\mathbb {R})\rightarrow L_{2}(\mathbb {R})$, given by

$$\begin{aligned} \widehat{f}(\xi )=(\mathcal {F}f)(\xi ):=\int _{\mathbb {R}} f(x)\,\mathrm {e}^{-2\pi \mathrm {i}x\xi }\,\mathrm {d}x. \end{aligned}$$

The group $\mathbb {G}$ acts on $L_2(\mathbb {R})$ by

$$\begin{aligned} \psi _\zeta (x)=\frac{1}{\sqrt{u}}\,\psi \left( \frac{x-v}{u}\right) . \end{aligned}$$

On the other hand, in the time-frequency case, $\mathbb {X}$ is the plane $\mathbb {R}^2=\{\zeta =(q,p);\ q,p\in \mathbb {R}\}$ with Lebesgue measure $\mathrm {d}\zeta =\mathrm {d}q\,\mathrm {d}p$ and $\Psi =\phi$ is an admissible window such that $\Vert \phi \Vert _{L_2(\mathbb {R})}=1$. The action of $\mathbb {R}^2$ on $L_2(\mathbb {R})$ is defined by

$$\begin{aligned} \phi _\zeta (x) = \mathrm {e}^{2\pi \mathrm {i}px} \phi (x-q). \end{aligned}$$

In what follows, $\Psi$ always means an admissible wavelet $\psi$ or an admissible window $\phi$.

Toeplitz localization operators For a fixed $\Psi$, the transform $W_{\Psi }: L_2(\mathbb {R})\rightarrow L_{2}(\mathbb {X},\,\mathrm {d}\zeta )$ defined by $(W_{\Psi } f)(\zeta )=\langle f,\Psi _{\zeta }\rangle$ is a linear isometry. Its image $W_{\Psi }(L_{2}(\mathbb {R}))$ is a closed subspace of the Hilbert space $L_{2}(\mathbb {X},\,\mathrm {d}\zeta )$, and is called the space of wavelet transforms, or short-time Fourier transforms, respectively. Function $(\eta ,\zeta )\mapsto \langle \Psi _\eta ,\Psi _\zeta \rangle$ is the reproducing kernel of the space $W_{\Psi }(L_{2}(\mathbb {R}))$, and the orthogonal projection from $L_{2}(\mathbb {X},\,\mathrm {d}\zeta )$ onto $W_{\Psi }(L_{2}(\mathbb {R}))$ can be expressed as the integral operator

$$\begin{aligned} (P_{\Psi }F)(\eta )=\int _{\mathbb {X}} F(\zeta ) \langle \Psi _\eta , \Psi _\zeta \rangle \,\mathrm {d}\zeta . \end{aligned}$$

For a given bounded function a on $\mathbb {X}$, we define the Toeplitz localization operator (TLO) $T_a^{\Psi }: W_\Psi (L_2(\mathbb {R}))\rightarrow W_\Psi (L_2(\mathbb {R}))$ with generating symbol a by $T_a^\Psi =P_{\Psi }M_{a}$, where $M_a$ is the operator of pointwise multiplication by a on $L_2(\mathbb {X},\mathrm {d}\zeta )$. Alternatively, TLO $T_{a}^\Psi$ may be viewed as acting on $L_2(\mathbb {R})$, given by

$$\begin{aligned} T_{a}^\Psi f = \int _{\mathbb {X}} a(\zeta ) (W_\Psi f)(\zeta ) \Psi _{\zeta }\,\mathrm {d}\zeta , \end{aligned}$$

where the integral is interpreted in a weak sense. In this form it resembles the reproducing formula (1), i.e., $T_1^\Psi f = f$.

Spectral functions as convolutions In this paper we restrict our attention to the generating symbols depending on the first coordinate in the phase space. As shown in [17], for each admissible $\Psi$, there exists a unitary operator $R_{\Psi }$ that diagonalizes each TLO $T_a^\Psi$ acting on $W_{\Psi }(L_{2}(\mathbb {R}))$ with a bounded symbol a on $\mathbb {R}$. That is,

$$\begin{aligned} R_\Psi T_a^\Psi R_\Psi ^* = \gamma ^\Psi _{a} I \end{aligned}$$

with an explicitly stated spectral function $\gamma ^\Psi _a: \mathbb {R}\rightarrow \mathbb {C}$. On the one hand, for the time-frequency case with the generating symbol $a\in L_\infty (\mathbb {R})$, the spectral function takes the form

$$\begin{aligned} \gamma _a^\phi (\xi ) = \int _{\mathbb {R}} a(q)\,|\phi (\xi -q)|^2\,\mathrm {d}q = (K_\phi *a)(\xi ), \quad \xi \in \mathbb {R}, \end{aligned}$$

where $*$ is the usual (additive) convolution on $\mathbb {R}$, and $K_\phi (z)=|\phi (z)|^2$ is the (time-frequency case) kernel function. On the other hand, for the wavelet case with the generating symbol $b\in L_\infty (\mathbb {R}_+)$, the spectral function was obtained in the following form:

$$\begin{aligned} \breve{\gamma }_b^\psi (x) = \int _{\mathbb {R}_+} b(u)\,|\widehat{\psi }(ux)|^2\,\frac{\mathrm {d}u}{u} = \left( \breve{K}_\psi \star \widetilde{b}\right) (x), \quad x\in \mathbb {R}, \end{aligned}$$

where $\star$ is the Mellin (multiplicative) convolution on $\mathbb {R}_+$ given by

$$\begin{aligned} (f\star g)(x):= \int _{\mathbb {R}_+} f(u)\,g\left( \frac{x}{u}\right) \,\frac{\mathrm {d}u}{u} \end{aligned}$$

whenever $f,g\in L_1\left( \mathbb {R}_+, \frac{\mathrm {d}u}{u}\right)$. Moreover, $\widetilde{b}(u):=b(1/u)$ for $b: \mathbb {R}_+\rightarrow \mathbb {C}$ is an involution, and $\breve{K}_\psi (z)=|\widehat{\psi }(z)|^2$ is the (wavelet case) kernel function.

The relationship between the additive and multiplicative convolution may be described via the uniform homeomorphisms $\eta (u)=-\ln u$ and $\theta (x)=\ln x$. Thus, given a function $a\in L_\infty (\mathbb {R})$, we transform it to $b=a\circ \eta \in L_\infty (\mathbb {R}_+)$, calculate the wavelet case spectral function $\breve{\gamma }_{b}^\psi$ and compose it with $\theta ^{-1}$. The resulting function $\gamma _{a}^\psi : \mathbb {R}\rightarrow \mathbb {C}$ has the form

$$\begin{aligned} \gamma _{a}^\psi (\xi )=\breve{\gamma }_{a\circ \eta }^\psi \left( \theta ^{-1}(\xi )\right) . \end{aligned}$$

In other words, after the changes of variables we can consider the generating symbols from $L_\infty (\mathbb {R})$ and the spectral functions defined on $\mathbb {R}$. Then $a\mapsto \gamma _{a}^\psi$ is a mapping connecting the original a with its transformed spectral function,

$$\begin{aligned} \gamma _{a}^\psi (\xi ) =\bigl (K_\psi *a\bigr )(\xi ) = \int _{\mathbb {R}} a(x)\, |\widehat{\psi }(\mathrm {e}^{\xi -x})|^2\,\mathrm {d}x, \quad \xi \in \mathbb {R}, \end{aligned}$$

where the convolution kernel $K_\psi$ is defined on $\mathbb {R}$ by $K_\psi = \breve{K}_\psi \circ \theta ^{-1}$. In this paper, when working with $a\in L_\infty (\mathbb {R})$, we always mean the transformed generating symbol in the case of TLOs related to wavelets.

In the following proposition we summarize the results of this section for both cases (time-frequency as well as wavelet) of spectral functions.

Proposition 3

Let $a\in L_\infty (\mathbb {R})$. Then $\gamma ^\Psi _{a}$ can be expressed as $\gamma ^\Psi _{a} = K_\Psi *a$. Consequently, each $\gamma ^\Psi _{a}$ belongs to the C$^*$-algebra $C_u(\mathbb {R})$ of uniformly continuous functions on $\mathbb {R}$.

3 Proof of Theorem 1: Approximate Deconvolution on the Real Line

In this section we describe a technique that provides a sufficient condition for density of the set $\mathfrak {G}_\Psi$ in terms of a convolution kernel $K_\Psi$ (expressed in terms of admissible wavelets/windows $\Psi$). In order to prove Theorem 1, we use the following concept of a Dirac sequence, see [23, XI, § 1].

Definition 4

A sequence $(f_n)_{n=1}^\infty$ of functions $f_n: \mathbb {R}\rightarrow [0,+\infty )$ is called a Dirac sequence, if it satisfies the following conditions:

• (a) $\int _{\mathbb {R}} f_n(x)\,\mathrm {d}x = 1$ for each $n\in \mathbb {N}$;

• (b) for each $\delta >0$ it holds

  $$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\mathbb {R}{\setminus } [-\delta , \delta ]} f_n(x)\,\mathrm {d} x = 0. \end{aligned}$$

The Dirac sequences form a subclass of approximate identities in the non-unital Banach algebra $L_1(\mathbb {R})$, and the following proposition implies that the Wiener’s condition ($\widehat{K}$ does not vanish) is sufficient for the approximate invertibility of K in $L_1(\mathbb {R})$.

Proposition 5

(approximate deconvolution on the real line) Suppose that $K\in L_1(\mathbb {R})$, such that $\widehat{K}$ does not vanish. Then there exists a sequence $(\phi _n)_{n=1}^\infty$ in $L_1(\mathbb {R})$, such that $(K*\phi _n)_{n=1}^\infty$ is a Dirac sequence.

Proof

Let $(\psi _n)_{n=1}^\infty$ be the Fejér kernel on the real line:

$$\begin{aligned} \psi _n(x)=\frac{\sin (n\pi x)^2}{n \pi ^2 x^2}. \end{aligned}$$

It is well known (see, for example, [26, subsection 2.3.7]) that $(\psi _n)_{n=1}^\infty$ is a Dirac sequence in $L_1(\mathbb {R})$. Additionally, the Fourier transforms of the functions $\psi _n$ have compact supports:

$$\begin{aligned} \widehat{\psi }_n(t)= {\left\{ \begin{array}{ll} 1-\frac{|t|}{n}, &{}\quad |t|\le n;\\ 0, &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Therefore, from Wiener’s Division Lemma (see, for example, [29, Lemma 1.4.2]), for every $n\in \mathbb {N}$ there exists a function $\phi _n\in L_1(\mathbb {R})$, such that $K*\phi _n=\psi _n$. $\square$

The following lemma is a well-known result for uniformly continuous functions, see, for example, [26, Theorem 2.2.21].

Lemma 6

Let $f\in C_u(\mathbb {R})$. If $(h_n)_{n=1}^\infty$ is a Dirac sequence, then $\lim \nolimits _{n\rightarrow \infty } \Vert f*h_n - f\Vert _\infty = 0$.

Proposition 7

Suppose that $K\in L_1(\mathbb {R})$, such that $\widehat{K}$ does not vanish on $\mathbb {R}$. Then the set $\mathfrak {S}=\{K*a;\;a\in L_\infty (\mathbb {R})\}$ is a dense subset of $C_u(\mathbb {R})$.

Proof

It is well-known that the convolution of a bounded function with an integrable function is bounded and uniformly continuous. Therefore, $\mathfrak {S}$ is contained in $C_u(\mathbb {R})$.

Let $\sigma \in C_u(\mathbb {R})$, and choose sequences $(\phi _n)_{n=1}^\infty$ and $(\psi _n)_{n=1}^\infty$ as in Proposition 5. We may define a sequence of functions $(f_n)_{n=1}^\infty$ by $f_n=\phi _n*\sigma$, and then

$$\begin{aligned} K *f_n = K *\phi _n *\sigma = \psi _n *\sigma . \end{aligned}$$

Since $(\psi _n)_{n=1}^\infty$ is a Dirac sequence, $\psi _n*\sigma$ converges uniformly to $\sigma$ by Lemma 6. $\square$

Proof of Theorem 1

Let $a\in L_\infty (\mathbb {R})$. By Proposition 3 we have $\gamma _a^\Psi = K_\Psi *a$. If $\Psi$ is such that $K_\Psi \in L_1(\mathbb {R})$ and the Fourier transform of $K_\Psi$ does not vanish on $\mathbb {R}$, then by Proposition 7 the set $\mathfrak {G}_\Psi =\{K_\Psi *a;\,a\in L_\infty (\mathbb {R})\}$ is a dense subset of $C_u(\mathbb {R})$. It follows that the C$^*$-algebra $\mathcal {T}_\Psi$ generated by TLOs with symbols $a\in L_\infty (\mathbb {R})$ is isomorphic to the C$^*$-algebra $C_u(\mathbb {R})$ and coincides with the closure of the set of its initial generators. $\square$

The following simple result states that, for a particular case, functions of the form $K*a$ are not only uniformly continuous, but also Lipschitz continuous. Let ${\text {AC}}(\mathbb {R})$ be the algebra of absolutely continuous functions on $\mathbb {R}$. Then $f\in {\text {AC}}(\mathbb {R})$ iff f has a derivative almost everywhere and $f'\in L_1(\mathbb {R})$. In particular, the condition of Proposition 8 holds if K belongs to the Schwartz class.

Proposition 8

Let $K\in L_1(\mathbb {R})\cap {\text {AC}}(\mathbb {R})$. Then, for every $a\in L_\infty (\mathbb {R})$, the function $K*a$ is Lipschitz continuous, i.e.,

$$\begin{aligned} |(K*a)(x_1)-(K*a)(x_2)| \le \Vert K'\Vert _1\,\Vert a\Vert _\infty \,|x_1-x_2|\qquad (x_1,x_2\in \mathbb {R}). \end{aligned}$$

Proof

We bound a by $\Vert a\Vert _\infty$,

$$\begin{aligned} |(K*a)(x_1)-(K*a)(x_2)| \le \Vert a\Vert _\infty \int _{\mathbb {R}} |K(x_1-y)-K(x_2-y)|\,\mathrm {d}y. \end{aligned}$$

Without loss of generality, suppose that $x_1<x_2$. Applying the second fundamental theorem of calculus and the Tonelli–Fubini theorem, we obtain

$$\begin{aligned} \int _{\mathbb {R}} |K(x_1-y)-K(x_2-y)|\,\mathrm {d}y&= \int _{\mathbb {R}} \left| \,\int _{x_1-y}^{x_2-y}K'(u)\,\mathrm {d}u\right| \,\mathrm {d}y\\&\quad \le \int _{\mathbb {R}} \int _{x_1-y}^{x_2-y} |K'(u)|\,\mathrm {d}u\,\mathrm {d}y \\&=\int _{\mathbb {R}} \int _{x_1-u}^{x_2-u} |K'(u)|\,\mathrm {d}y\,\mathrm {d}u =(x_2-x_1)\Vert K'\Vert _1, \end{aligned}$$

which completes the proof. $\square$

Note that if the convolution kernel $K_\Psi$ belongs to $L_1(\mathbb {R})\cap {\text {AC}}(\mathbb {R})$, then by Proposition 8 the set $\mathfrak {G}_\Psi$ is a proper subset of $C_u(\mathbb {R})$, because there are bounded uniformly continuous functions on $\mathbb {R}$ that are not Lipschitz continuous.

4 Proof of Theorem 2: a Vertical True Poly-Analytic Case Study

In this section we apply the general density result from previous section to an important special case of TLOs related to wavelets from Laguerre functions. In what follows we recall necessary notions and constructions from [18].

Let $\mathbb {G}=\{\zeta =(u,v); \,\,u>0, v\in \mathbb {R}\}$ be the affine group with left-invariant Haar measure $\mathrm {d}\nu _L(\zeta )=u^{-2}\,\mathrm {d}u\mathrm {d}v$. We identify $\mathbb {G}$ with the upper half-plane $\Pi =\{\zeta =v+\mathrm {i}u;\,v\in \mathbb {R}, u>0\}$ in the complex plane $\mathbb {C}$, with hyperbolic metric and corresponding (hyperbolic) measure $\mathrm {d}\nu _L$. Then $L_2(\mathbb {G},\mathrm {d}\nu _L)$ is the Hilbert space of all square-integrable complex-valued functions on $\mathbb {G}$ with respect to the measure $\mathrm {d}\nu _L$.

For $k\in \mathbb {Z}_+$, consider the parameterized family of admissible wavelets $\psi ^{(k)}$ on $\mathbb {R}$ defined on the Fourier transform side as

$$\begin{aligned} \widehat{\psi }^{(k)}(\xi )=\chi _+(\xi )\sqrt{2\xi }\,\ell _k(2\xi ), \end{aligned}$$

where $\ell _n(x):=\mathrm {e}^{-x/2} L_n(x)$ are the (simple) Laguerre functions, where $L_n(x)$ is the Laguerre polynomial of order $n\in \mathbb {Z}_+$; and $\chi _+$ is the characteristic function of the positive half-line. Then, for a fixed $k\in \mathbb {Z}_+$, the space $W_{\psi ^{(k)}}(L_2(\mathbb {R}))$ is isometrically isomorphic to

$$\begin{aligned} A^{(k)} := \left\{ W_k f(u,v); \,f\in H_2(\mathbb {R})\right\} \subset L_2(\mathbb {G},\mathrm {d}\nu _L), \end{aligned}$$

where $W_k f$ is the continuous wavelet transforms of Hardy-space functions $f\in H_2(\mathbb {R})$ with respect to wavelets $\psi ^{(k)}$. Importantly, for $k\in \mathbb {N}$, the space $A^{(k)}$ coincides with the true poly-analytic Bergman space of order $k+1$ on the upper half-plane. For further details we refer the interested reader to [16] and [28].

Given a generating symbol $a = a(\zeta )$, $\zeta \in \mathbb {G}$, the Toeplitz operator $T_a^{\psi ^{(k)}}$ acting on $W_{\psi ^{(k)}}(L_2(\mathbb {R}))$ is usually called the Calderón-Toeplitz operator (CTO), see [25, 30]. From the diagonalization described in Sect. 2, each CTO $T_a^{\psi ^{(k)}}$ with a vertical symbol $a=a(u)$ on $\mathbb {G}$ is unitarily equivalent to the multiplication operator acting on $L_2(\mathbb {R}_+)$ with the spectral function $\gamma _a^{\psi ^{(k)}}$. For simplicity, we write $T_a^{(k)}$ instead of $T_a^{\psi ^{(k)}}$, and $\gamma _{a,k}$ instead of $\gamma _a^{\psi ^{(k)}}$. The explicit form of spectral function $\gamma _{a,k}: \mathbb {R}_+\rightarrow \mathbb {C}$ is as follows:

$$\begin{aligned} \gamma _{a,k}(u) = \int _{\mathbb {R}_+} a\left( \frac{v}{2u}\right) \ell _k^2(v)\,\mathrm {d}v, \end{aligned}$$

see [16, Theorem 3.2]. We denote the set $\{\gamma _{a,k};\,a\in L_\infty (\mathbb {R}_+)\}$ by $\mathfrak {G}_k$. Recall that in the case $k=0$, corresponding to Toeplitz operators with bounded vertical symbols acting on the non-weighted Bergman space of analytic functions over the upper half-plane, the diagonalization (i.e., the reduction of $T_a^{(0)}$ to $\gamma _{a,0}$) was found already by Vasilevski [32]. Immediately, it follows that the C$^*$-algebra generated by CTOs $T_a^{(0)}$ with bounded vertical symbols a is commutative and is isometrically isomorphic to the C$^*$-algebra generated by the set $\mathfrak {G}_0$. It was proved in [12] that the set $\mathfrak {G}_0$ is dense in the C$^*$-algebra ${\text {VSO}}(\mathbb {R}_+)$. In the case $k = 0$ the corresponding convolution kernel is somewhat simpler, and an approximately inverse sequence $(\phi _n)_{n=1}^\infty$ was constructed explicitly in [12]. For a general $k\in \mathbb {Z}_+$ we were not able to construct an approximately inverse sequence $(\phi _n)_{n=1}^\infty$ in such an explicit manner. Instead of that, we apply Theorem 1, which means that we use a much less explicit construction of $\phi _n$ explained in the proof of Proposition 5. Thus, the present proof of Theorem 2 is divided into four steps:

• I. We transform the density problem to the real line via an appropriate change of variables to allow Wiener’s deconvolution technique to be applied.

• II. We express the Fourier transform of the convolution kernel as a product of the gamma function with a complex polynomial $P_k$.

• III. To prove that $P_k$ has no real zero, we express $P_k$ as a linear combination of symmetric Meixner-Pollaczek polynomials and move the zeros to the real line using certain linear transformations.

• IV. We show that the transformed polynomials satisfy a three-term recurrence relation, thus being orthogonal. This verifies Wiener’s sufficiency condition $\widehat{K_k}(t)\ne 0$ for the set $\mathfrak {G}_k$ being a dense set in ${\text {VSO}}(\mathbb {R}_+)$ for each $k\in \mathbb {Z}_+$.

4.1 Step I: Transforming the Problem to the Real Line

After the change of variables $u=\mathrm {e}^x/2$, $v=\mathrm {e}^y$, with

$$\begin{aligned} b(x)=a(\mathrm {e}^{-x})\quad \text {and}\quad \Gamma _{b,k}(x)=\gamma _{a,k}(\mathrm {e}^x/2), \end{aligned}$$

we obtain

$$\begin{aligned} \Gamma _{b,k}(x) = \int _{\mathbb {R}} b(x-y) K_k(y)\,\mathrm {d}y, \end{aligned}$$

where

$$\begin{aligned} K_k(x) = \mathrm {e}^x \ell _k^2(\mathrm {e}^x) = \frac{\mathrm {e}^x}{\mathrm {e}^{\mathrm {e}^x}} (L_k(\mathrm {e}^x))^2. \end{aligned}$$

Note that $K_k\in L_1(\mathbb {R})$ for each k, and

$$\begin{aligned} \Vert K_k\Vert _1 = \int _\mathbb {R}K_k(x)\,\mathrm {d}x = \int _\mathbb {R}\mathrm {e}^x \ell _k^2(\mathrm {e}^x)\,\mathrm {d}x = \int _{\mathbb {R}_+} \ell _k^2(t)\,\mathrm {d}t = 1. \end{aligned}$$

We aim to prove that, for each $k\in \mathbb {Z}_+$, the set

$$\begin{aligned} \mathfrak {S}_k = \{\Gamma _{b,k};\,b\in L_\infty (\mathbb {R})\} \end{aligned}$$

is dense in the C$^*$-algebra $C_u(\mathbb {R})$. Since $b\in L_\infty (\mathbb {R})$ and $K_k\in L_1(\mathbb {R})$, each function $\Gamma _{b,k}$ belongs to $C_u(\mathbb {R})$. From Theorem 1, it is sufficient to prove that the Fourier transform of $K_k$ does not vanish on $\mathbb {R}$, i.e.,

$$\begin{aligned} (\forall t\in \mathbb {R})\quad \widehat{K}_k(t)\ne 0. \end{aligned}$$

4.2 Step II: Fourier Transform of the Convolution Kernel

Let us start with Howell’s formula [14], and also [7, formula 8.976($3^6$)],

$$\begin{aligned} \left( L_n^{(\lambda )}(x)\right) ^2= \frac{\Gamma (1+\lambda +n)}{2^{2n} n!}\sum _{r=0}^{n} \left( {\begin{array}{c}2n-2r\\ n-r\end{array}}\right) \frac{(2r)!}{r!}\frac{1}{\Gamma (1+\lambda +r)} L_{2r}^{(2\lambda )}(2x). \end{aligned}$$

In particular, for $\lambda =0$,

$$\begin{aligned} L_k^2(x) = \frac{1}{2^{2k}}\sum _{r=0}^{k} \alpha _r(k) L_{2r}(2x), \quad \text {where}\quad \alpha _r(k):= \left( {\begin{array}{c}2k-2r\\ k-r\end{array}}\right) \left( {\begin{array}{c}2r\\ r\end{array}}\right) . \end{aligned}$$

Then the convolution kernel $K_k$ can be written as

$$\begin{aligned} K_k(x)=\frac{1}{2^{2k}}\frac{\mathrm {e}^x}{\mathrm {e}^{\mathrm {e}^x}}\;\sum _{r=0}^{k} \alpha _r(k)\,L_{2r}(2\mathrm {e}^{x}), \end{aligned}$$

and the Fourier transform of each summand is

$$\begin{aligned} \int _{\mathbb {R}}L_{2r}(2\mathrm {e}^x)\,\mathrm {e}^{-2\pi {\text {i}}x t-\mathrm {e}^x}\,\mathrm {e}^x\mathrm {d}x&= 2^{2\pi {\text {i}}t -1} \int _{\mathbb {R}_+} L_{2r}(z)\,\mathrm {e}^{-z/2} z^{-2\pi {\text {i}}t}\,\mathrm {d}z. \end{aligned}$$

Applying [7, formula 7.414(7)] for the last integral,

$$\begin{aligned} 2^{2\pi \mathrm {i}t -1} \int _{\mathbb {R}_+} L_{2r}(z)\,\mathrm {e}^{-z/2} z^{-2\pi {\text {i}}t}\,\mathrm {d}z&= \Gamma (1-2\pi {\text {i}}t) \phantom {.}_2F_1(-2r,1-2\pi {\text {i}}t; 1;2) \\&= \Gamma (1-2\pi {\text {i}}t) \phantom {.}_2F_1(-2r,2\pi {\text {i}}t; 1;2), \end{aligned}$$

where $\phantom {.}_2F_1(a,b;c;z)$ is the Gaussian hypergeometric function. Thus, the Fourier transform of $K_k$ is

$$\begin{aligned} \widehat{K}_k(t) = \frac{(-1)^k}{(k!)^2}\,\Gamma (1-2\pi {\text {i}}t)P_k(2\pi t), \end{aligned}$$

where

$$\begin{aligned} P_k(t) = \frac{(-1)^k\;(k!)^2}{2^{2k}}\;\sum _{r=0}^{k} \alpha _r(k)\phantom {.}_2F_1(-2r,{\text {i}}t; 1;2). \end{aligned}$$

The first of these polynomials are

$$\begin{aligned} P_0(t)&= 1, \\ P_1(t)&= -1 + {\text {i}}t + t^2,\\ P_2(t)&= 4 - 6 {\text {i}}t - 7 t^2 + 2 {\text {i}}t^3 + t^4,\quad \text {and}\\ P_3(t)&= -36 + 66 {\text {i}}t + 85 t^2 - 39 {\text {i}}t^3 - 22 t^4 + 3 {\text {i}}t^5 + t^6. \end{aligned}$$

Since $\Gamma$ has no zeros, it remains to show that each polynomial $P_k$ has no real root, i.e., $P_k(t)\ne 0$ for every $t\in \mathbb {R}$.

4.3 Step III: $P_k$ as a Linear Combination of Orthogonal Polynomials

Meixner polynomials of the second kind $m^{(\lambda )}_n(x,\phi )$ (also known as Meixner-Pollaczek polynomials) are [21, formula (9.7.1)]

$$\begin{aligned} m_n^{(\lambda )}(x,\phi ):=\frac{(2\lambda )_n}{n!}\,\mathrm {e}^{{\text {i}}n\phi } \phantom {.}_2F_1\left( -n,\lambda +{\text {i}}x; 2\lambda ;1-\mathrm {e}^{-2{\text {i}}\phi }\right) . \end{aligned}$$

Since

$$\begin{aligned} \phantom {.}_2F_1(-2r,{\text {i}}t; 1;2) = (-1)^r\, m_{2r}^{(1/2)} \left( t+\frac{{\text {i}}}{2}, \frac{\pi }{2}\right) , \end{aligned}$$

each $P_k$ can be written as

$$\begin{aligned} P_k(t) = \frac{(-1)^k (k!)^2}{2^{2k}} \sum _{r=0}^k (-1)^r\,\alpha _r(k)\, m_{2r}^{(1/2)}\left( t+\frac{{\text {i}}}{2},\frac{\pi }{2}\right) . \end{aligned}$$

Some numerical experiments in Wolfram Mathematica show that the zeros of $P_k$ lie on the horizontal line ${\text {Im}}t=-1/2$. This observation motivates the following change of variables:

$$\begin{aligned} Q_k(x) = 4^k\, P_k\left( \frac{x-{\text {i}}}{2}\right) . \end{aligned}$$

Substituting the expression for $P_k$ yields the following form:

$$\begin{aligned} Q_k(x) = (-1)^k (k!)^2\, \sum _{r=0}^{k} (-1)^r\alpha _r(k)\, m_{2r}^{(1/2)}\left( \frac{x}{2},\frac{\pi }{2}\right) . \end{aligned}$$

The first of these polynomials are

$$\begin{aligned} Q_0(x)&= 1, \\ Q_1(x)&= -3 + x^2,\\ Q_2(x)&= 41 - 22 x^2 + x^4, \,\,\,\text {and}\\ Q_3(x)&= -1323 + 907x^2 - 73 x^4 + x^6. \end{aligned}$$

The polynomials $m_n^{(1/2)}(\frac{x}{2},\frac{\pi }{2})$ are of special interest because they possess many interesting properties, see [1]. They are known as the symmetric Meixner-Pollaczek polynomials, because they satisfy the symmetric relation

$$\begin{aligned} m_n^{(1/2)}\left( \frac{x}{2},\frac{\pi }{2}\right) =(-1)^n\, m_n^{(1/2)}\left( -\frac{x}{2},\frac{\pi }{2}\right) , \end{aligned}$$

which means that each polynomial $m_{2r}^{(1/2)}(\frac{x}{2},\frac{\pi }{2})$ is an even function. Thus, each polynomial $Q_k$ is also an even function.

Let $M_n$ be the (transformed) monic form of $m_n^{(1/2)}(\frac{x}{2},\frac{\pi }{2})$, i.e.,

$$\begin{aligned} M_n(x) :=(2n)!\, m_{2n}^{(1/2)}\left( \frac{\sqrt{x}}{2},\frac{\pi }{2}\right) . \end{aligned}$$

These monic forms are completely described by the recurrence formula [1]

$$\begin{aligned} x M_n(x) = M_{n+1}(x) + a_n M_n(x) + b_n M_{n-1}(x), \quad n\in \mathbb {Z}_+, \end{aligned}$$

(2)

where $M_{-1}(x)=0$, $M_0(x)=1$, and $a_n = 4n(2n+1)+1$, $b_n = 4n^2(2n-1)^2$. Then we write $Q_k(x)$ as $R_k(x^2)$, where

$$\begin{aligned} R_k(x) = \sum _{r=0}^{k} \beta _r(k) M_{r}(x) \quad \text {with}\quad \beta _r(k):=(-1)^{k-r}\left( {\begin{array}{c}k\\ r\end{array}}\right) ^2(2k-2r)! \end{aligned}$$

The first of these polynomials are

$$\begin{aligned} R_0(x)&= 1, \\ R_1(x)&= -3 + x,\\ R_2(x)&= 41 - 22 x + x^2, \,\,\,\text {and}\\ R_3(x)&= -1323 + 907x - 73 x^2 + x^3. \end{aligned}$$

Thus, each $R_k$ is a monic polynomial of degree k and $R_k$ alternates its sign (because $M_k$ alternates sign).

Remark 9

The appearance of the symmetric Meixner-Pollaczek polynomials on the Fourier transform side of the convolution kernel related to Laguerre functions is quite natural, because the Laguerre functions and Meixner-Pollaczek polynomials are connected via the Mellin transform, see [20, 22].

4.4 Step IV: $(R_k)_{k=0}^{\infty }$ is a Family of Orthogonal Polynomials

Proposition 10

The polynomials $R_k$ satisfy the recurrence relation

$$\begin{aligned} xR_{k}(x) = R_{k+1}(x) + A_k R_{k}(x) + B_k R_{k-1}(x) \end{aligned}$$

(3)

with $A_k=8k(k+1) + 3$, $B_k = 16 k^4$, and $R_{-1}(x)=0$, $R_0(x)=1$.

Proof

For $k=0$ the result is trivial. Consider $k\ge 1$. From

$$\begin{aligned} xR_{k}(x) = x M_k(x) + \sum _{r=0}^{k-1} \beta _r(k) xM_{r}(x) \end{aligned}$$

and the recurrence relation (2) for polynomials $M_r$, it follows that

$$\begin{aligned} xR_{k}(x)&= R_{k+1}(x)+A_kR_{k}(x)+B_kR_{k-1}(x) \\&\quad + [a_k+\beta _{k-1}(k)-\beta _{k}(k+1)-A_k]M_{k}(x) \\&\quad + \sum _{r=0}^{k-1}[b_{r+1}\beta _{r+1}(k)+(a_r-A_k)\beta _{r}(k)\\&\quad +\beta _{r-1}(k)-\beta _{r}(k+1)-B_k\beta _{r}(k-1)]M_r(x), \end{aligned}$$

using the convention $\beta _{-1}(k):=0$. Then $R_{k}$ satisfies the three-term recurrence relation if and only if

$$\begin{aligned} a_k+\beta _{k-1}(k)-\beta _{k}(k+1)&= A_k, \\ b_{r+1}\beta _{r+1}(k)+(a_r-A_k)\beta _{r}(k)-B_k\beta _{r}(k-1)&= \beta _{r}(k+1)-\beta _{r-1}(k) \end{aligned}$$

for $r=0,1,\ldots ,k-1$. The first identity holds since $\beta _n(n+1)=-2(n+1)^2$ for each n, and so

$$\begin{aligned} a_k+\beta _{k-1}(k)-\beta _k(k+1) = 4k(2k+1)+1-2k^2+2(k+1)^2 = 8k^2+8k+3 = A_k. \end{aligned}$$

For the second identities we apply some binomial identities. For each $k\ge 1$ and $r=0,1,\ldots ,k-1$, we have

$$\begin{aligned} \beta _{r}(k+1)-\beta _{r-1}(k)&= (-1)^{k+r+1}(2k-2r+2)!\left( \left( {\begin{array}{c}k+1\\ r\end{array}}\right) ^2-\left( {\begin{array}{c}k\\ r-1\end{array}}\right) ^2\right) \\&= (-1)^{k+r+1}(2k-2r+2)!\left( {\begin{array}{c}k\\ r\end{array}}\right) \left( {\begin{array}{c}k+1\\ r\end{array}}\right) \frac{k+1+r}{k+1} \\&= \lambda _r(k)\cdot (2k-2r+1)(2k-2r-1)(k+r+1), \end{aligned}$$

where Pascal’s identity,

$$\begin{aligned} \left( {\begin{array}{c}n+1\\ m\end{array}}\right) -\left( {\begin{array}{c}n\\ m-1\end{array}}\right) = \left( {\begin{array}{c}n\\ m\end{array}}\right) , \end{aligned}$$

as well as the binomial coefficient identities,

$$\begin{aligned} \left( {\begin{array}{c}n+1\\ m\end{array}}\right) +\left( {\begin{array}{c}n\\ m-1\end{array}}\right)&= \left( {\begin{array}{c}n+1\\ m\end{array}}\right) \frac{n+1+m}{n+1}, \\ (m+1)\left( {\begin{array}{c}n\\ m+1\end{array}}\right)&= n\left( {\begin{array}{c}n-1\\ m\end{array}}\right) , \end{aligned}$$

have been used, and

$$\begin{aligned} \lambda _r(k) := 4\,(-1)^{r+k+1}\left( {\begin{array}{c}k\\ r\end{array}}\right) ^2(k-r)\cdot (2k-2r+2)!. \end{aligned}$$

On the other hand, the same manipulation with binomial coefficient identities yields

$$\begin{aligned} b_{r+1}\beta _{r+1}(k)&= \lambda _r(k)\cdot (2r+1)^2(k-r), \\ (a_r-A_k)\beta _{r}(k)&= \lambda _r(k)\cdot (2k-2r-1)[4k(k+1)-2r(2r+1)+1], \\ B_k\beta _{r}(k-1)&= \lambda _r(k)\cdot 4k^2(k-r). \end{aligned}$$

Thus,

$$\begin{aligned} b_{r+1}\beta _{r+1}(k)+(a_r-A_k)\beta _{r}(k)-B_k\beta _{r}(k-1) = \lambda _r(k) \cdot \Theta (k,r), \end{aligned}$$

where

$$\begin{aligned} \Theta (k,r){:=} (2r+1)^2(k-r)+(2k-2r-1)[4k(k+1)-2r(2r+1)+1] - 4k^2(k-r). \end{aligned}$$

Easy algebraic computations then yield

$$\begin{aligned} \Theta (k,r)=(2k-2r+1)(2k-2r-1)(k+r+1). \end{aligned}$$

Thus, the sequence $(R_k)_{k=0}^\infty$ satisfies the recurrence relation (3). $\square$

Final arguments From Favard’s theorem [24], the polynomials $R_k$ are orthogonal with respect to a positive measure on $\mathbb {R}$. Therefore, all zeros of $R_k$ are real and simple. Thus, the zeros of $Q_k$ belong to $\mathbb {R}\cup {\text {i}}\mathbb {R}$, and the zeros of $P_k$ belong to

$$\begin{aligned} \left( \mathbb {R}-\frac{1}{2}{\text {i}}\right) \cup \left( {\text {i}}\mathbb {R}-\frac{1}{2}{\text {i}}\right) . \end{aligned}$$

The unique intersection of this set with the real line is the point 0, and it can be verified (using a routine manipulation with binomial coefficients) that

$$\begin{aligned} P_k(0)=(-1)^k (k!)^2 \ne 0. \end{aligned}$$

Thus, each polynomial $P_k$ does not vanish on $\mathbb {R}$, and it follows that

$$\begin{aligned} \widehat{K}_k(t) = \frac{(-1)^k}{(k!)^2}\,\Gamma (1-2\pi {\text {i}}t)P_k(2\pi t) \end{aligned}$$

also does not vanish on $\mathbb {R}$. Therefore, from Theorem 1, the set $\mathfrak {S}_k =\{\Gamma _{b,k}; b\in L_\infty (\mathbb {R})\}$ is dense in $C_u(\mathbb {R})$ and this implies that the set $\mathfrak {G}_k$ is dense in ${\text {VSO}}(\mathbb {R}_+)$ for each $k\in \mathbb {Z}_+$. The proof of Theorem 2 is complete.

Remark 11

Let $\mathfrak {T}_k$ be the set of Calderón-Toeplitz operators $T_a^{(k)}$ acting on $A^{(k)}$ with bounded vertical symbols a. From Huang [15], the von Neumann algebra $W^*(\mathfrak {T}_k)$ generated by $\mathfrak {T}_k$ is maximal, and $W^*(\mathfrak {T}_k)$ is the closure of the C$^*$-algebra $\mathcal {T}_k$ generated by $\mathfrak {T}_k$ with respect to the strong operator topology in the algebra $\mathcal {B}\left( A^{(k)}\right)$ of all linear bounded operators on the true poly-analytic Bergman space $A^{(k)}$ over the upper half-plane. These outcomes do not depend on the value of $k\in \mathbb {Z}_+$. For a description of C$^*$-algebras of Toeplitz operators with vertical symbols acting on each poly-analytic Bergman spaces (not only true poly-analytic ones) over the upper half-plane we refer the interested reader to recent paper [28].