Abstract
We describe some radial Fock type spaces which possess Riesz bases of normalized reproducing kernels, the spaces \(\mathcal F_{\varphi }\) of entire functions f such that \(fe^{-\varphi }\in L_2(\mathbb C)\), where \(\varphi (z) = \varphi (|z|)\) is a radial subharmonic function. We prove that \(\mathcal F_{\varphi }\) has Riesz basis of normalized reproducing kernels for sufficiently regular \(\psi (r)=\varphi (e^r)\) such that \(\psi ''(r)\) is bounded above.
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1 Introduction
We consider the radial weighted Fock spaces
where dm(z) being planar Lebesgue measure, \(\varphi (z)\) being a radial subharmonic function. We assume that this space is not degenerate. It has a natural Hilbert space structure, the evaluations \(\delta _\lambda : f\rightarrow f(\lambda )\) are continuous. Since the Hilbert spaces are self-dual, it follows that each of these functionals is generated by an element \(k_\lambda (z)=k(z,\lambda )\in \mathcal F_\varphi \) in the sense that
The function \(k(z,\lambda )\) is called the reproducing kernel of the space \(\mathcal F_\varphi \). Obviously,
The system \(\{ k(z,\lambda _j)\}_{j=1}^{\infty }\) will be called an unconditional basis in the space \(\mathcal F_\varphi \) if it is complete and for some \(C>1\) we have
for finite sequences \(\{a_j\}\) of complex numbers. An unconditional basis \(\{e_j,\ j= 1,2,...\}\) becomes Riesz basis if and only if \(0<\inf \nolimits _k \Vert e_k\Vert \le \sup \nolimits _k \Vert e_k\Vert <\infty \). Equivalently, Riesz basis is a linear isomorphic image of an orthonormal basis in a separable Hilbert space. We study the existence of Riesz bases of normalized reproducing kernels \(\left\{ \frac{k(z,\lambda _j)}{\Vert k(\cdot ,\lambda _j)\Vert }\right\} _{j=1}^{\infty }\) in \(\mathcal F_\varphi \).
The issue on existence and construction of Riesz bases of normalized reproducing kernels is actively studied due to the fact, in particular, that this question is closely related to such classical problems of complex analysis as the problem of interpolation (see, for example, [1,2,3]) and the problem of representing by exponential series (see, for example, [4]). Summing up the studies of this issue in various aspects, we can say that Riesz bases are a rare phenomenon (see [1, 3, 5]). In [5], an unexpected result was obtained, which stated the existence of Riesz bases of normalized reproducing kernels in the Fock spaces \(\mathcal F_\varphi \) with the weights \(\varphi =(\ln ^+|z|)^\alpha \) as \(\alpha \in (1;2]\). Later, in paper [6], there was proved the existence of Riesz bases of normalized reproducing kernels in the Fock spaces with radial weights of essentially more general form. We prove that if \(\varphi \) is a radial function and the function \(\psi (r)=\varphi (e^r)\) satisfies the conditions: \(\lim _{r\rightarrow \infty } \psi '(r)=\infty \), \(\psi ''\) is a non-increasing positive function, and \(\left| \psi '''(r) \right| =O(\psi ''(r)^{\frac{5}{3}})\), \(r\rightarrow \infty \), then \(\mathcal F_\varphi \) has a Riesz basis of normalized reproducing kernels. In this paper, we prove a weaker sufficient condition for the existence of a Riesz basis of normalized reproducing kernels in Fock spaces with radial and sufficiently regular weights.
2 Notation, definitions, preliminaries, and statements of results
Definition 1
A convex function v is called regular if there exist a number \(q>1\) and a function \(\gamma (x) \uparrow +\infty \) such that
Conditions of this kind are used to find the asymptotic of the Laplace integrals. In this paper we prove (see Theorem 3) that if \(\varphi \) is radial subharmonic function, the function \(\psi (x)=\varphi (e^x)\) is regular, and
then the space \(\mathcal F_\varphi \) has a Riesz basis of normalized reproducing kernels.
Definition 2
The function \( \widetilde{v}(y)=\sup \limits _x(xy-v(x)), \ y\in \mathbb R, \) is the Young conjugate of the convex function v.
Definition 3
Let v be a continuous function, and
We set
for a positive number p.
This characteristic was introduced in [7].
Definition 4
Let v be a convex function on \(\mathbb R\), and p be a positive number. We let
where \(v'_+\) is the right derivative of v.
This characteristic was introduced in [8]. It was proved in [7] (see Lemma 3) that
for convex function v.
In what follows we shall make use of the following notations. For positive functions \(A,\ B,\) the writing \(A(x)\asymp B(x)\), \(x\in X\), means that for some constants \(C,\ c>0\) and for all \(x\in X\) the estimates \(cB(x)\le A(x) \le CB(x)\) hold. The symbol \(A(x)\prec B(x)\), \(x\in X\), (\(A(x)\succ B(x)\), \(x\in X\)), means the existence of a constant \(C>0\) such that \(A(x) \le CB(x)\) (\(B(x) \le CA(x)\)).
We denote [x] the floor function (the integer part of x).
3 A sufficient condition for the existence of Riesz bases in general Hilbert spaces
In this section, we consider a sufficient condition for the existence of Riesz bases of normalized reproducing kernels in general Hilbert spaces of entire functions. Let H be a radial functional Hilbert space of entire functions satisfying the division property, i.e.:
-
1.
all evaluation functionals \(\delta _z \, : \, f\rightarrow f(z)\) are continuous;
-
2.
if \(F\in H\), then \(\Vert F\Vert = \Vert F(ze^{i\varphi })\Vert \) for any \(\varphi \in \mathbb R\);
-
3.
if \(F\in H\), \(F(z_0)=0\), then \(F(z)(z-z_0)^{-1}\in H\).
The functional property of the space implies that it admits a reproducing kernel \(k( z , \lambda )\).
It was proved in [9] (see Theorem A) that if H is a radial functional Hilbert space satisfying the division property, admitting a Riesz basis of normalized reproducing kernels, and monomials are complete in H, then there exists a convex sequence u(n), \(n\in \mathbb N\cup \{ 0\}\), such that \(\Vert z\Vert ^n\asymp e^{u(n)}\), \(n\in \mathbb N\cup \{ 0\}\). The convexity of \(\{ u(n)\}\) means
If u(t) be a convex piecewise linear function with integer non-negative breakpoints, and \(u(t)\equiv u(0)\) as \(t<0\), then the convexity condition can be written in a more compact form
In what follows, we assume that \(u(n)=\ln \Vert z^n\Vert \), \(n\in \mathbb N\cup \{0\}\), is a convex sequence, \(u(0)=0\), and u(t) is a piecewise linear function, \(u(t)\equiv 0\) as \(t<0\). The following theorem was proved in [10] (see Theorem 2).
Theorem A
If the system of monomials \( \left\{ z^n, \, n\in \mathbb N\cup \{0\}\right\} \) is complete in a radial functional Hilbert space H satisfying the division property, and the function \(\widetilde{u}\) satisfies the condition
then the space H possesses Riesz bases of normalized reproducing kernels.
Let us prove following lemmas.
Lemma 1
For the convex piecewise linear function u(t), \(t\in \mathbb R\), condition (2) is equivalent to
Proof
Without loss of generality we can suppose that \(N\ge 1\) in (2). The monotonicity of the function \(\widetilde{u}_+'(x)\) implies that if (2) holds, then
By definition of \(\rho _2(\widetilde{u} ,x,1)\) this means that
Thus (3) holds.
Conversely, let
By definition of \(\rho _2(\widetilde{u} ,x,1)\) we have
and therefore,
Let \(N=\left[ \frac{1}{\delta }\right] +1\). Taking into account that \(\widetilde{u}_+'\) is an increasing function, we get
that is, (2) holds. \(\square \)
Lemma 2
Condition (3) is equivalent to the boundness of the function \(\rho _2(u,t,1)\) on \(\mathbb R_+\):
Proof
Let \(\rho _2( u,t,1)\le N,\ t\in \mathbb R_+,\) for some constant \(N>0\). Without loss of generality we can suppose that N is integer. By definition of \(\rho _2( u,t,1)\) this means that
Hence, since \(u_+'(y)\) is a monotonic function, we have
or
It was proved in [11] (see Lemma 2) that the the Young conjugate \(\widetilde{u}\) is also piecewise linear with breakpoints \(x_n=u_+'(n-1)=u(n)-u(n-1)\), and the derivative \(\widetilde{u}_+'\) is the function with unite jumps at the points \(x_n\). Thus, the last estimate can be written as
This means that the quantity of jumps of \(\widetilde{u}_+'\) on an interval which length is less than \(\frac{1}{2N}\) does not exceed 2N. Since there are unit jumps, we find that for \(\varepsilon <\frac{1}{2N}\)
Put \(\varepsilon =\frac{1}{5N}\). Then
Hence,
Conversely, let for some \(\varepsilon >0\)
Then
Hence, for any \(x\ge 1\)
Put \(N=\left[ \frac{1}{\varepsilon }\right] \). Then
or
Thus,
Hence,
It was proved in [7] (see Lemmas 3 and 4) that the function \(\rho _2(u ,x,1)\) satisfies Lipschitz condition
Therefore,
\(\square \)
Now we can reformulate Theorem A in the following form.
Theorem 1
If the system of monomials \( \left\{ z^n, \, n\in \mathbb N\cup \{0\}\right\} \) is complete in a radial functional Hilbert space H satisfying the division property, and the function u satisfies the condition
then the space H possesses Riesz bases of normalized reproducing kernels.
4 A sufficient condition for the existence of Riesz bases in radial weighted Fock spaces in terms of conjugate function
Let us turn to Fock spaces with radial weight \(\varphi \). Let \(\psi (x)=\varphi (e^x)\) and
Then \(u_1(t)\) is a convex function on \(\mathbb R_+\), coinciding with the function u(t) at the points \(t\in \mathbb N\cup \{ 0\}\), in particular,
Let us extend \(u_1\) to the entire axis, setting \(u_1(t)\equiv 0\), \(t\in \mathbb R_-\).
Lemma 3
We have the relation
Proof
Let
Let us suppose that for a natural number n, satisfying \(|n-t|\le \frac{1}{2}\), the following inequality holds
Then, setting \(k=[M]+1\), we have
that is
Since the functions u and \(u_1\) coincide at integer points, then
Hence,
and
The resulting contradiction means that
Since the function \(\rho _2(u,t,1)\) satisfies the Lipschitz condition, we have
The second relation is proved in a similar way. \(\square \)
Lemma 4
If the function \(\widetilde{\psi }\) is regular and q is the constant in the regularity condition, then for sufficiently large numbers \(t\in \mathbb R\) the following inequalities hold
Proof
Let \(\rho _0=\frac{\gamma (t)}{\sqrt{\widetilde{\psi }''(t)}} \), then due to regularity \(\widetilde{\psi }\)
Hence, by the mean value theorem for any x such that \(|x-t|\le \rho _0\) we have
Therefore, if \(\gamma (t)\ge q\), then
Hence, \(\rho _2(\widetilde{\psi },t,1):=\rho \le \rho _0\). By definition of the function \(\rho _2(\widetilde{\psi },t,1)\) we have
and
From this and the regularity of the function \(\widetilde{\psi }\), we obtain the assertion of the lemma for t such that \(\gamma (t)\ge q\). \(\square \)
Lemma 5
If the function \(\widetilde{\psi }\) is regular, then for some constant \(m>1\) we have
The left estimate holds without the regularity condition.
Proof
1. Let us prove the left inequality. By Theorem 2(a) in [7] we have
That is,
for some \(a>0\), and
Take an arbitrary point \(t\in \mathbb R_+\) and denote \(\rho _1(\widetilde{\psi },t+1,1)=\rho _1\). Let \(\alpha \in \left( 0;\frac{1}{2}\right) \). There is a linear function l(x) such that
For the linear function \(l_1(x)=l(x)-\frac{1}{2}\ln \rho _1\) we have
The function \(\rho _1 (u,x,p)\) satisfies the Lipschitz condition too (see Lemma 4 in [7]), therefore, if \(|x-t|\le \alpha \rho _1\), then
or
Hence,
Continuing estimate (4), we obtain
and by that,
or taking into account the arbitrariness of \(\alpha \in \left( 0;\frac{1}{2}\right) \), we get
By (1) we get
Hence, by Lemma 2 in [7] we obtain the lower estimate with the constant \(m=2(2a+3)\).
2. Let \(\widetilde{\psi }\) be a regular function. It is convenient to write the regularity condition in the form
where \(\gamma _1(x)=\gamma (x+1)\). By Theorem 2(a) in [7] we have
that is, for some \(b>0\) we have
or
By Lemma 4 we have
and by the regularity of \(\widetilde{\psi }\) we get
Take a point \(t\in \mathbb R_+\) so that \(\gamma _1(t)>3\sqrt{q}(b+\ln q +1)\) and denote \(\rho _2(\widetilde{\psi },t+1,1)=\rho _2\). Let \(c=\ln q\). Then by the last estimate and by (5) we obtain
or
Suppose that
Then there is a linear function l(x) such that
By Lemma 4 we have
Therefore, by (6), taking into account the choice of t, for the linear function \(l_1(x)=l(x)+\frac{1}{2} \ln \rho _2\), we obtain
for \(|x-t|\le 3\sqrt{q}(b+c+1)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}\). Hence,
and by (1),
Then by Lemma 4 we get
Hence, by Lemma 2 in [7] we obtain
or
Since \(\rho _2(\widetilde{\psi }, t+1,1)>0\), we obtain a contradiction. Thus,
Taking into account (1) again, for t such that \(\gamma _1(t)\ge 3\sqrt{q}(b+c+1)\) we have
Since the functions \(\rho _2(u_1,t,1)\) and \(\rho _2(\widetilde{\psi }, t,1)\) are continuous, this implies the estimate
for some constant \(A>0\). \(\square \)
Lemmas 3–5 imply the following theorem.
Theorem 2
If \(\widetilde{\psi }\) is a regular function, and \(\widetilde{\psi }''(t)\) satisfies the condition
then the Fock space with the weight \(\psi (\ln |z|)\) possesses Riesz bases of normalized reproducing kernels.
5 A sufficient condition for the existence of Riesz bases in radial weighted Fock spaces in terms of weight
In this section we will prove the final theorem.
Theorem 3
If \(\psi \) is a regular function, and
then the Fock space with the weight \(\psi (\ln |z|)\) possesses Riesz bases of normalized reproducing kernels.
Let us first prove a lemma.
Lemma 6
Let \(v\in C^2(\mathbb R)\) be a convex indefinitely increasing function which is not linear on \(\mathbb R_+\). If v is a regular function, then the conjugate function \(\widetilde{v}\) is also regular on some interval \((a;+\infty )\).
Proof
By hypothesis of the lemma, \(v'\) is strictly increasing, and we have
Let \(x_\pm =x\pm \frac{1}{2}\frac{\gamma (x)}{\sqrt{v''(x)}}\) and \(t=v'(x), \ t_\pm =v'(x_\pm )\). Let us note that
By regularity of v, for some \(x^*\in [x_-;x_+]\) we have
Let \(\tau =\frac{1}{2}\left( t_++t_-\right) .\) Then \(y=\widetilde{v}'(\tau )\in [x_-;x_+]\), and since \(t_-\ge 0\), \(\tau \ge \frac{1}{2}t_+\), then by (8) we get
If
then by (7) we get
that is \(s\in [t_-;t_+]\) and \(\widetilde{v}'(s)\in [x_-;x_+]\). Hence, by (7) we obtain
Thus, \(\widetilde{v}\) satisfies the regularity condition at the points \(\tau (x)\) with the function \(\frac{\gamma (x(\tau ))}{2q^{\frac{3}{2}}}\). By (9), the set of such \(\tau \) contains some interval \((a;+\infty )\). On this interval, the regularity condition will also hold with the increasing function
\(\square \)
By Lemma 6 and by (7) we obtain that if the hypothesis of Theorem 3 is satisfied then the function \(\widetilde{\psi }\) is regular and \(\inf _{t>0}\widetilde{\psi }''(t)>0\). Then by Theorem 2, the Fock space with the weight \(\psi (\ln |z|)\) possesses Riesz bases of normalized reproducing kernels.
Corollary 1
f \(\psi \in C^2\) and \(0<\psi ''(t)\asymp 1,\ t\in \mathbb R\), then the Fock space with the weight \(\psi (\ln |\lambda |)\) possesses Riesz bases of normalized reproducing kernels.
Proof
In this case, the conditions of Theorem 3 are satisfied in an obvious way. \(\square \)
Corollary 2
If \(\psi \in C^3\), \(\psi '(x)\) is unlimited and
then the Fock space with the weight \(\psi (\ln |\lambda |)\) possesses Riesz bases of normalized reproducing kernels.
Proof
By (7) we obtain that
and for some \(M>0\) we have
Hence, by mean value theorem we have
Let
Then \(\gamma (x)\uparrow +\infty \) as \(x \rightarrow +\infty \), and \(\gamma (x)\le \ln \widetilde{\psi }''(x)\). Put
then \(C<\infty \). If
then \(|x-y|\le C\). Hence, by (10) we have
Thus, \(\widetilde{\psi }(x)\) is regular with \(q=e^{MC}\). By Lemma 6\(\psi (x)\) is regular too. By Theorem 3 the Fock space with the weight \(\psi (\ln |\lambda |)\) possesses Riesz bases of normalized reproducing kernels. \(\square \)
Note that the Corollaries 1 and 2 are close to [6, Theorem 1.2]. It proved the existence of unconditional bases provided that the nonincreasing function \(\psi ''(t)\) satisfies the condition
Monotonicity implies the existence of a limit
If \(\psi _0>0\), then the condition of the Corollary 1 is satisfied and the other conditions of Theorem 1.2 are not needed. If \(\psi _0=0\), then we get the situation of the Corollary 2 without monotonicity and with a weaker condition for \(\psi '''\).
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Isaev, K.P., Yulmukhametov, R.S. Riesz bases of normalized reproducing kernels in Fock type spaces. Anal.Math.Phys. 12, 11 (2022). https://doi.org/10.1007/s13324-021-00623-z
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DOI: https://doi.org/10.1007/s13324-021-00623-z