The Young type theorem in weighted Fock spaces

TL;DR

This paper generalizes a theorem about the completeness of biorthogonal systems in weighted Fock spaces, extending classical results from Paley-Wiener and Bargmann-Fock spaces under mild regularity conditions.

Contribution

It proves that in any radial weighted Fock space, the biorthogonal system to a complete, minimal set of kernels is also complete, broadening the scope of Young's theorem.

Findings

Biorthogonal systems are complete in weighted Fock spaces.

Generalization of Young's theorem to a broader class of spaces.

Extension of classical results to weighted Fock space settings.

Abstract

We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley--Wiener space and a recent result of Belov for the classical Bargmann--Segal--Fock space.

Equations

$$f(\lambda )=⟨f,k_{\lambda }{⟩}_{\mathcal{H}},f\in \mathcal{H}.$$

$$g_{\lambda }(z)=\frac{G(z)}{G^{\prime }(\lambda )(z-\lambda )}$$

$$f\sim \sum _{\lambda \in \Lambda }⟨f,k_{\lambda }⟩g_{\lambda }=\sum _{\lambda \in \Lambda }f(\lambda )\frac{G}{G^{\prime }(\lambda )(\cdot -\lambda )},$$

$$\mathcal{F}_{h}=\Big{\{}f\in\operatorname{Hol}({\mathbb{C}}):\|f\|^{2}_{\mathcal{F}_{h}}=\int_{{\mathbb{C}}}|f(z)|^{2}e^{-h(z)}\,dm_{2}(z)<\infty\Big{\}},$$

$$\log (t+|h^{\prime }(t)|+|h^{\prime \prime }(t)|)=o(h(t)),t\to \infty .$$

$$|h(w)-h(z)|=o(1),w\in D(z,e^{-\epsilon h(z)}),|z|\to \infty .$$

$$\delta \le |h(s)-h(t)|\le |s-t|\cdot |h^{\prime }(s^{\prime })|\le C\exp (-\epsilon h(t))\exp ((\epsilon /2)h(t)),$$

$$|f(z)|+|f^{\prime }(z)|\le C_{\epsilon }{e}^{((1/2)+\epsilon )h(z)}\Vert f{\Vert }_{{\mathcal{F}}_h},f\in {\mathcal{F}}_h,z\in \mathbb{C}.$$

$$\psi (w)=h(z+\gamma w),w\in \mathbb{D}.$$

$$\|\nabla\psi(w)\|+|\Delta\psi(w)|\\ \lesssim\gamma|h^{\prime}(|z+\gamma w|)|+\gamma^{2}\Bigl{(}|h^{\prime\prime}(|z+\gamma w|)|+\frac{|h^{\prime}(|z+\gamma w|)|}{|z+\gamma w|}\Bigr{)}=o(1)$$

$$\rho(z)=\int_{{\mathbb{D}}}\log\Bigl{|}\frac{z-w}{1-\bar{w}z}\Bigr{|}\Delta\psi(w)\,dm_{2}(w),\qquad z\in{\mathbb{D}}.$$

$$|\rho (w)|+\Vert \nabla \rho (w)\Vert =o(1),w\in \mathbb{D},|z|\to \infty ,$$

$$F(w)=f(z+\gamma w)e^{-({\psi }_1(w)+i\widetilde{{\psi }_1}(w))/2},w\in \mathbb{D}.$$

$${\int }_{\mathbb{D}}|F(w){|}^{2}d{m}_2(w)\lesssim {\gamma }^{-2}{\int }_{D(z,\gamma )}|f(\zeta ){|}^2{e}^{-h(\zeta )}d{m}_2(\zeta )\lesssim {\gamma }^{-2}\Vert f{\Vert }_{{\mathcal{F}}_h}^2.$$

$$|f(z)|$$

$$|f^{\prime }(z)|$$

$$\lesssim {\gamma }^{-2}{e}^{h(z)/2}\Vert f{\Vert }_{{\mathcal{F}}_h}.$$

$$\zeta \mapsto \frac{G(z)P(\zeta )-G(\zeta )P(z)}{z-\zeta }$$

$$Q(z)={\int }_{\mathbb{C}}\frac{G(z)P(\zeta )-G(\zeta )P(z)}{z-\zeta }\overline{H(\zeta )}{e}^{-h(\zeta )}d{m}_2(\zeta ),z\in \mathbb{C}.$$

$$Q=GT$$

$$Q(w_j)={\int }_{\mathbb{C}}\frac{G(w_j)P(\zeta )}{w_j-\zeta }\overline{H(\zeta )}{e}^{-h(\zeta )}d{m}_2(\zeta ).$$

$$\begin{gathered} Q(z)=G(z){\int }_{\mathbb{C}}\frac{P(\zeta )\overline{H(\zeta )}}{z-\zeta }{e}^{-h(\zeta )}d{m}_2(\zeta ) \\ -P(z){\int }_{\mathbb{C}}\frac{G(\zeta )\overline{H(\zeta )}}{z-\zeta }{e}^{-h(\zeta )}d{m}_2(\zeta ) \\ \stackrel{\mathrm{def}}{=}{A}_1(z)+A_2(z). \end{gathered}$$

$$|\phi (z)|+\Vert \nabla \phi (z)\Vert \lesssim (1+|z|{)}^{-3},z\in \mathbb{C}.$$

$$\Bigl{|}\int_{\mathbb{C}}\frac{\varphi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}\lesssim(1+|z|)^{-1},\qquad z\in{\mathbb{C}}.$$

$$\Bigl{|}\int_{\mathbb{C}}\frac{\psi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}\lesssim e^{\alpha h(z)},\qquad z\in{\mathbb{C}}.$$

$$\displaystyle\Bigl{|}\int_{\mathbb{C}}\frac{\varphi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}\leq\Bigl{|}\int_{D(0,|z|/2)}\frac{\varphi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}+\Bigl{|}\int_{D(z,1)}\frac{\varphi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}$$

$$\displaystyle+\Bigl{|}\int_{{\mathbb{C}}\setminus(D(0,|z|/2)\cup D(z,1))}\frac{\varphi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}$$

$$\displaystyle\lesssim\frac{1}{1+|z|}+\int_{D(z,1)}\Bigl{|}\frac{\varphi(\zeta)-\varphi(z)}{z-\zeta}\Bigr{|}\,dm_{2}(\zeta)+\int_{|z|/2}\frac{dt}{t^{2}}\lesssim\frac{1}{1+|z|}.$$

$$\Bigl{|}\int_{\mathbb{C}}\frac{\psi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}\\ \leq\Bigl{|}\int_{{\mathbb{C}}\setminus D(z,\eta)}\frac{\psi(\zeta)\,dm_{2}(\zeta)}{z-\zeta}\Bigr{|}+\Bigl{|}\int_{D(z,\eta)}\frac{\psi(\zeta)-\psi(z)}{z-\zeta}\,dm_{2}(\zeta)\Bigr{|}\\ \lesssim e^{\alpha h(z)}+1\lesssim e^{\alpha h(z)}.$$

$$\Big{|}G(z)\int_{\mathbb{C}}\frac{P(\zeta)\overline{H(\zeta)}}{z-\zeta}\,e^{-h(\zeta)}\,dm_{2}(\zeta)\Big{|}\lesssim\frac{|G(z)|}{1+|z|}.$$

$$\Big{|}\int_{\mathbb{C}}\frac{\psi(z)}{z-\zeta}\,dm_{2}(\zeta)\Big{|}\lesssim e^{\alpha h(z)},\qquad z\in\mathbb{C},$$

$$\begin{gathered} {\int }_{\mathbb{C}}\frac{G(\zeta )\overline{H(\zeta )}}{z-\zeta }{e}^{-h(\zeta )}d{m}_2(\zeta ) \\ =(z-\lambda ){\int }_{\mathbb{C}}\frac{G(\zeta )\overline{H(\zeta )}}{(\zeta -\lambda )(z-\zeta )}{e}^{-h(\zeta )}d{m}_2(\zeta )-{\int }_{\mathbb{C}}\frac{G(\zeta )\overline{H(\zeta )}}{\zeta -\lambda }{e}^{-h(\zeta )}d{m}_2(\zeta ) \\ =(z-\lambda ){\int }_{\mathbb{C}}\frac{\psi }{z-\cdot }d{m}_2-{\int }_{\mathbb{C}}\psi d{m}_2. \end{gathered}$$

Full text

The work was supported by Russian Science Foundation grant 14-41-00010.

1. Introduction and the main result

Let $\mathcal{H}$ be a reproducing kernel Hilbert space whose elements are entire functions. This means that the evaluation functionals $f\mapsto f(\lambda)$ are continuous on $\mathcal{H}$ for every $\lambda\in\mathbb{C}$, and so there exist functions $k_{\lambda}\in\mathcal{H}$ (reproducing kernels) such that

[TABLE]

It is well known that the structure properties of $\mathcal{H}$ are closely related to geometric properties of systems of reproducing kernels. E.g., one of the simplest relations of this type says that a set $\Lambda=\{\lambda_{n}\}$ is a uniqueness set for $\mathcal{H}$ if and only if the system of reproducing kernels $\{k_{\lambda}\}_{\lambda\in\Lambda}$ is complete in $\mathcal{H}$.

We say that the space $\mathcal{H}$ has the division property if for any $f\in\mathcal{H}$ and any $w\in\mathbb{C}$ such that $f(w)=0$, we have $\frac{f(z)}{z-w}\in\mathcal{H}$. Most of the classical spaces of entire functions (Paley–Wiener, de Branges, Fock spaces) possess this property.

Now assume that a system $\{k_{\lambda}\}_{\lambda\in\Lambda}$ is complete and minimal in $\mathcal{H}$ (i.e., $k_{\lambda}$ does not belong to the closed linear span of $\{k_{\mu}\}_{\mu\in\Lambda\setminus\{\lambda\}}$, $\lambda\in\Lambda$). Fix $\lambda_{0}\in\Lambda$ and let $g$ be a function in $\mathcal{H}$ which is orthogonal to $\{k_{\mu}\}_{\mu\in\Lambda\setminus\{\lambda_{0}\}}$ and so vanishes on $\Lambda\setminus\{\lambda_{0}\}$. Then the function $G=(\cdot-\lambda_{0})g$ (known as the generating function of the sequence $\{k_{\lambda}\}_{\lambda\in\Lambda}$ and unique up to a multiplicative constant) vanishes on $\Lambda$ and, by the division property, $\frac{G(z)}{z-\lambda}\in\mathcal{H}$ for any $\lambda\in\Lambda$. Clearly, the functions

[TABLE]

form a system biorthogonal to $\{k_{\lambda}\}_{\lambda\in\Lambda}$, i.e., $\langle k_{\lambda},g_{\mu}\rangle=\delta_{\lambda\mu}$, $\lambda,\mu\in\Lambda$.

Given any function $f\in\mathcal{H}$, consider its (formal) Fourier series with respect to the biorthogonal pair $\{k_{\lambda}\}_{\lambda\in\Lambda}$, $\{g_{\lambda}\}_{\lambda\in\Lambda}$:

[TABLE]

which is nothing else than the Lagrange-type interpolation series for $f$. The possibility of reconstruction of a function $f$ (in a given class) from its Lagrange series (1.2) is one of the basic problems of function theory. In this note we address one special instance of this problem, namely the question of completeness of the biorthogonal system $\{g_{\lambda}\}_{\lambda\in\Lambda}$. Clearly, this is a necessary (though by no means sufficient) condition for any reasonable reconstruction method. More delicate necessary conditions were studied in [1, 2].

It is well known and easy to see that in general a system biorthogonal to a complete and minimal system in a Hilbert space need not to be complete. For special systems of functions the situation may be different. R. Young [7] showed that every complete and minimal system of exponentials $\{e^{i\lambda t}\}$ in $L^{2}(-\pi,\pi)$ (or, equivalently, any complete and minimal system of reproducing kernels in the Paley–Wiener space $PW_{\pi}$) has a complete biorthogonal system. This result was also obtained independently and by different methods by G. Gubreev and A. Kovalenko [6]. In [3] the first two authors studied completeness of biorthogonal systems in the context of de Branges spaces (or model subspaces of the Hardy space) and showed that in these spaces there exist complete and minimal systems of reproducing kernels whose biorthogonal system is incomplete with any given (even infinite) defect.

Recently, Yu. Belov proved a Young-type result for the classical Bargmann–Segal–Fock space $\mathcal{F}$, see the definition below. The aim of this note is to extend this result to all radial weighted Fock spaces satisfying very mild restrictions on the weight. In particular, our method provides a simple proof for the classical Fock space $\mathcal{F}$. This method can also be easily modified to work in other reproducing kernel Hilbert spaces of entire functions (e.g., we show how to deduce the result of Young using this method).

Now we introduce the radial weighted Fock spaces. Let $h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a continuous function.

Extend $h$ to $\mathbb{C}$ by $h(z):=h(|z|)$, $z\in\mathbb{C}$. Consider the class of functions

[TABLE]

$dm_{2}$ being planar Lebesgue measure. Each space $\mathcal{F}_{h}$ is a reproducing kernel Hilbert space with respect to the norm $\|\cdot\|_{\mathcal{F}_{h}}$. The classical Fock (or Bargmann–Segal–Fock) space corresponds to $h(r)=\pi r^{2}$.

Our main result is the following theorem:

Theorem 1**.**

Let $h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a function in $C^{2}(\mathbb{R}_{+})$ such that

[TABLE]

Let $\{k_{\lambda}\}_{\lambda\in\Lambda}$ be a complete and minimal system of reproducing kernels in $\mathcal{F}_{h}$. Then the system $\{g_{\lambda}\}_{\lambda\in\Lambda}$ biorthogonal to the system $\{k_{\lambda}\}_{\lambda\in\Lambda}$ is complete in $\mathcal{F}_{h}$.

We do not consider the case of finite dimensional Fock spaces $\mathcal{F}_{h}$, $\log(t)\not=o(h(t))$, $t\to\infty$.

2. Preliminaries

In this section we establish some standard growth estimates for the elements of the space $\mathcal{F}_{h}$. In what follows we denote by $D(z,r)$ the open disc with center $z$ of radius $r$ and we write $\mathbb{D}$ instead of $D(0,1)$.

Lemma 2**.**

Given ${\varepsilon}>0$, we have

[TABLE]

Proof.

Fix $\delta>0$. For large $t>0$ denote by $s$ a point in the interval $[t-\exp(-{\varepsilon}h(t)),t+\exp(-{\varepsilon}h(t))]$ closest to $t$ and such that $|h(s)-h(t)|\geq\delta$ (if such points exist at all; otherwise, there is nothing to prove).

Then by (1.3) for some point $s^{\prime}$ between $s$ and $t$ we have

[TABLE]

which is impossible for large $t$. ∎

Lemma 3**.**

Given ${\varepsilon}>0$ there exists $C_{\varepsilon}>0$ such that

[TABLE]

Proof.

(Compare to the proof of Lemma 4.1 in [5].) Given $z\in{\mathbb{C}}$, $\beta>0$, set $\gamma=\exp(-\beta h(z))$,

[TABLE]

Then by (1.3),

[TABLE]

uniformly in $w\in{\mathbb{D}}$ when $|z|\to\infty$. Here and later on we write $a\lesssim b$ if $a\leq Cb$ for some constant $C$. Set

[TABLE]

Then

[TABLE]

and $\psi_{1}=\psi-\rho$ is real and harmonic on ${\mathbb{D}}$. Denote by $\widetilde{\psi_{1}}$ the harmonic conjugate of $\psi_{1}$ and set

[TABLE]

Then $F$ is analytic on ${\mathbb{D}}$ and

[TABLE]

Therefore,

[TABLE]

∎

3. Proof of Theorem 1

Fix a complete and minimal system $\{k_{\lambda}\}_{\lambda\in\Lambda}$ in $\mathcal{F}_{h}$. Its biorthogonal system is given by (1.1), where the generating function $G$ has simple zeros exactly at $\Lambda$. Suppose that the system $\{g_{\lambda}\}_{\lambda\in\Lambda}$ is not complete and fix $H\in\mathcal{F}_{h}\setminus\{0\}$ orthogonal to this system. Since polynomials are dense in any radial Fock space, we can choose $n\geq 0$ such that $\langle H,z^{n}\rangle\not=0$ and fix a polynomial $P$ of degree $n+1$ with simple zeros $w_{1},\ldots,w_{n+1}$ in ${\mathbb{C}}\setminus\Lambda$.

Given $z\in{\mathbb{C}}$, the function

[TABLE]

is in $\mathcal{F}_{h}$. Set

[TABLE]

Then $Q$ is an entire function vanishing at $\Lambda$, and hence,

[TABLE]

for an entire function $T$. For every zero $w_{j}$ of $P$ we have

[TABLE]

Therefore, if $T\equiv 0$, then $H$ is orthogonal to all $P/(w_{j}-\cdot)$, and hence to $z^{n}$, which is impossible. To complete the proof it remains to verify that $Q\in\mathcal{F}_{h}$.

We have

[TABLE]

The inclusion $Q\in\mathcal{F}_{h}$ will be a consequence of the following two statements.

Lemma 4**.**

Let $\varphi\in L^{1}({\mathbb{C}})\cap C^{1}({\mathbb{C}})$, $\|\varphi\|_{L^{1}({\mathbb{C}})}\lesssim 1$. Suppose that

[TABLE]

Then

[TABLE]

Lemma 5**.**

Let $\alpha>0$ and suppose that $\psi\in L^{1}({\mathbb{C}})\cap C^{1}({\mathbb{C}})$, $\|\psi\|_{L^{1}({\mathbb{C}})}\lesssim 1$, $\|\nabla\psi\|\lesssim e^{\alpha h}$ on $\mathbb{C}$. Then

[TABLE]

Proof of Lemma 4.

Let $z\in{\mathbb{C}}$, $|z|\geq 2$. We have

[TABLE]

∎

Proof of Lemma 5.

Let $z\in{\mathbb{C}}$. Set $\eta=e^{-\alpha h(z)}$. We have

[TABLE]

Here we use the fact that $\int_{D(z,r)}(z-\zeta)^{-1}dm_{2}(\zeta)=0$ for any disc $D(z,r)$. ∎

To complete the proof of Theorem 1, we apply Lemmas 4 and 5 and estimate $A_{1}$ and $A_{2}$ in the right hand side of (3.1). By Lemma 3, the function $\varphi=P\overline{H}e^{-h}$ satisfies the conditions of Lemma 4, and hence

[TABLE]

Since $G/(1+|\cdot|)\in L^{2}(e^{-h}dm_{2})$, we obtain that $A_{1}\in L^{2}(e^{-h}dm_{2})$.

Next, choose $\lambda\in\Lambda$ and set $\psi=G\overline{H}e^{-h}/(\cdot-\lambda)$. By Lemma 3, $\|\nabla\psi\|\lesssim e^{\alpha h}$ on $\mathbb{C}$ for every $\alpha>0$. Furthermore, $\psi\in L^{1}(\mathbb{C})$. Therefore, by Lemma 5,

[TABLE]

for every $\alpha>0$. Next,

[TABLE]

Thus,

[TABLE]

Finally, $Q\in\mathcal{F}_{h}$, which proves Theorem 1. ∎

Remark 6**.**

Let us show how to prove Young’s theorem on reproducing kernels in the Paley–Wiener space $PW_{\pi}$ (the Fourier image of $L^{2}(-\pi,\pi)$) using a similar idea. Note that $PW_{\pi}$ does not contain polynomials. Instead we use the fact that the family \big{\{}\frac{\sin\pi z}{\pi(z-n)}\big{\}}_{n\in\mathbb{Z}} is an orthonormal basis in $PW_{\pi}$. Set

[TABLE]

It is not too difficult to verify that the system \big{\{}\frac{S(z)}{z-\mu}\big{\}}_{\mu\in\mathcal{Z}} is complete in $PW_{\pi}$. Now assume that $\{k_{\lambda}\}_{\lambda\in\Lambda}$ is a complete and minimal system of reproducing kernels in $PW_{\pi}$ and $\{g_{\lambda}\}_{\lambda\in\Lambda}$ is its biorthogonal system given by (1.1). Without loss of generality we may assume that $\Lambda\cap\mathcal{Z}=\emptyset$. If $H$ is orthogonal to $\{g_{\lambda}\}_{\lambda\in\Lambda}$, we define

[TABLE]

As before, $Q$ vanishes on $\Lambda$. Using standard estimates for the functions in the Paley–Wiener space (e.g., the fact that $f$ and $f^{\prime}$ are bounded on $\mathbb{R}$ for $f\in PW_{\pi}$), we conclude that $Q\in PW_{\pi}$ and, hence, $Q\equiv 0$. Thus, $H$ is orthogonal to the system \big{\{}\frac{S(z)}{z-\mu}\big{\}}_{\mu\in\mathcal{Z}} and, hence, is zero. **