Riesz bases of normalized reproducing kernels in Fock type spaces

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.

1 Introduction

We consider the radial weighted Fock spaces

$$\begin{aligned} \mathcal F_\varphi =\left\{ f\in H(\mathbb C):\ \Vert f\Vert ^2=\frac{1}{2\pi }\int \limits _{\mathbb C}|f(z)|^2e^{-2\varphi (z)}dm(z)<\infty \right\} , \end{aligned}$$

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

$$\begin{aligned} \frac{1}{2\pi }\int \limits _{\mathbb C}f(z)\overline{k(z,\lambda )}e^{-2\varphi (z)}dm(z)=f(\lambda ),\quad \text{ for } \text{ any } f\in \mathcal F_\varphi \, \text{ and }\, \lambda \in \mathbb C. \end{aligned}$$

The function $k(z,\lambda )$ is called the reproducing kernel of the space $\mathcal F_\varphi$. Obviously,

$$\begin{aligned} \Vert \delta _\lambda \Vert ^2=k(\lambda ,\lambda ):=K(\lambda ),\quad \lambda \in \mathbb C. \end{aligned}$$

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

$$\begin{aligned} \frac{1}{C} \sum _j|a_j|^2K(\lambda _j)\le \left\| \sum _ja_jk(z,\lambda _j)\right\| ^2\le C \sum _j|a_j|^2K(\lambda _j), \end{aligned}$$

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

$$\begin{aligned} \frac{1}{q}\le \frac{v ''(x)}{v ''(y)}\le q \, \, \, \text{ as }\, \, \, |x-y|\le \gamma (x)\sqrt{\frac{1}{v ''(x)}}, \, \, \, x,y\in \mathbb R_+. \end{aligned}$$

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

$$\begin{aligned} \sup _{x>0}\psi ''(x)<\infty , \end{aligned}$$

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

$$\begin{aligned} d(v,y,r)=\inf _l \left\{ \max _{t\in [y-r;y+r]}|v(t)-l(t)|, \ l \ \text{ is } \text{ a } \text{ linear } \text{ function } \right\} . \end{aligned}$$

We set

$$\begin{aligned} \rho _1(v,y,p) =\sup \{ r:\ d(v,y,r)\le p\} \end{aligned}$$

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

$$\begin{aligned} \rho _2(v,x,p)=\sup \left\{ t>0:\ \int \limits _{x-t}^{x+t}\left| v'_+(\tau )-v'_+(x)\right| d\tau \le p\right\} , \end{aligned}$$

where $v'_+$ is the right derivative of v.

This characteristic was introduced in [8]. It was proved in [7] (see Lemma 3) that

$$\begin{aligned} \rho _1(v,y,p)=\rho _2(v,y,2p) \end{aligned}$$

(1)

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

$$\begin{aligned} u(n+1)+u(n-1)-2u(n)\ge 0,\quad n\in \mathbb N. \end{aligned}$$

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

$$\begin{aligned} u_+'(n+1)-u_+'(n)\ge 0. \end{aligned}$$

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

$$\begin{aligned} \sup \limits _{x>0}(\widetilde{u}'_+(x+1)-\widetilde{u}'_+(x))\le N<\infty , \end{aligned}$$

(2)

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

$$\begin{aligned} \inf _{x>1}\rho _2(\widetilde{u} ,x,1)>0. \end{aligned}$$

(3)

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

$$\begin{aligned} \int \limits _{x-\frac{1}{2N}}^{x+\frac{1}{2N}}|\widetilde{u}_+'(\tau )-\widetilde{u}_+'(x)|d\tau \le 1. \end{aligned}$$

By definition of $\rho _2(\widetilde{u} ,x,1)$ this means that

$$\begin{aligned} \rho _2(\widetilde{u} ,x,1)\ge \frac{1}{2N},\ x\ge 1. \end{aligned}$$

Thus (3) holds.

Conversely, let

$$\begin{aligned} \rho _2(\widetilde{u} ,x, 1)\ge 2\delta >0,\ x\ge 1. \end{aligned}$$

By definition of $\rho _2(\widetilde{u} ,x,1)$ we have

$$\begin{aligned} \int \limits _{\delta }^{2\delta }(\widetilde{u}_+'(x+t)-\widetilde{u}_+'(x))dt\le 1,\ x\ge 1, \end{aligned}$$

and therefore,

$$\begin{aligned} \widetilde{u}_+'(x+\delta )-\widetilde{u}_+'(x)\le \frac{1}{\delta },\ x\ge 1. \end{aligned}$$

Let $N=\left[ \frac{1}{\delta }\right] +1$. Taking into account that $\widetilde{u}_+'$ is an increasing function, we get

$$\begin{aligned} \widetilde{u}_+'(x+1)-\widetilde{u}_+'(x)\le \sum _{k=0}^{N-1}(\widetilde{u}_+'(x+k\delta +\delta )-\widetilde{u}_+'(x+\delta ))\le \frac{N}{\delta }\le N^2,\ x>0, \end{aligned}$$

that is, (2) holds. $\square$

Lemma 2

Condition (3) is equivalent to the boundness of the function $\rho _2(u,t,1)$ on $\mathbb R_+$:

$$\begin{aligned} \sup _{t>0}\rho _2(u,t,1)<\infty . \end{aligned}$$

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

$$\begin{aligned} \int \limits _{t-N}^{t+N}|u_+'(y)-u_+'(t)|dy\ge 1,\ t\in \mathbb R_+. \end{aligned}$$

Hence, since $u_+'(y)$ is a monotonic function, we have

$$\begin{aligned} u_+'(n+N)-u_+'(n-N)\ge \frac{1}{2N},\ n\in \mathbb N, \end{aligned}$$

or

$$\begin{aligned} u_+'(n+2N)-u_+'(n)\ge \frac{1}{2N},\ n\in \mathbb N. \end{aligned}$$

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

$$\begin{aligned} x_{n+2N}-x_n\ge \frac{1}{2N},\ n\in \mathbb N. \end{aligned}$$

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 $\varepsilon <\frac{1}{2N}$

$$\begin{aligned} \widetilde{u}_+'(x+\varepsilon )-\widetilde{u}_+'(x)\le 2N, \ x\ge 1. \end{aligned}$$

Put $\varepsilon =\frac{1}{5N}$. Then

$$\begin{aligned} \int \limits _{t-\varepsilon }^{t+\varepsilon }|\widetilde{u}_+'(x )-\widetilde{u}_+'(t)|dx\le 2N\cdot 2\varepsilon =\frac{4}{5}<1,\ t\ge 1. \end{aligned}$$

Hence,

$$\begin{aligned} \rho _2(\widetilde{u},t,1)\ge \frac{1}{5N},\ t\ge 1. \end{aligned}$$

Conversely, let for some $\varepsilon >0$

$$\begin{aligned} \rho _2(\widetilde{u},t,1)\ge 2\varepsilon ,\ t\ge 1. \end{aligned}$$

Then

$$\begin{aligned} \int \limits _{x+\varepsilon }^{x+2\varepsilon }|\widetilde{u}_+'(y)-\widetilde{u}_+'(x)|dy\le \int \limits _{x-2\varepsilon }^{x+2\varepsilon }|\widetilde{u}_+'(y)-\widetilde{u}_+'(x)|dy\le 1. \end{aligned}$$

Hence, for any $x\ge 1$

$$\begin{aligned} \widetilde{u}_+'(x+\varepsilon )-\widetilde{u}_+'(x)\le \frac{1}{\varepsilon }. \end{aligned}$$

Put $N=\left[ \frac{1}{\varepsilon }\right]$. Then

$$\begin{aligned} \widetilde{u}_+'(x+\varepsilon )-\widetilde{u}_+'(x)\le N+1, \end{aligned}$$

or

$$\begin{aligned} u_+'(n+N+1)-u_+'(n)\ge \varepsilon ,\ n\in \mathbb N\cup \{0\}. \end{aligned}$$

Thus,

$$\begin{aligned} \int \limits _{n+N+1}^{n+2(N+1)}|u_+'(x)-u_+'(n)|dx\ge \varepsilon (N+1)> 1. \end{aligned}$$

Hence,

$$\begin{aligned} \rho _2(u,n,1)\le 2N+2. \end{aligned}$$

It was proved in [7] (see Lemmas 3 and 4) that the function $\rho _2(u ,x,1)$ satisfies Lipschitz condition

$$\begin{aligned} |\rho _2(u ,x,1)-\rho _2(u ,y,1)|\le |x-y|,\ x,y\in \mathbb R. \end{aligned}$$

Therefore,

$$\begin{aligned} \rho _2(u,t,1)\le 2N+3,\ t\in \mathbb R_+. \end{aligned}$$

$\square$

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

$$\begin{aligned} \sup _{x>0}\rho _2(u,t,1)< \infty , \end{aligned}$$

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

$$\begin{aligned} e^{2u_1(t)}=\int \limits _{-\infty }^{\infty }e^{2(t+1)x-2\psi (x)}dx, \ t\in \mathbb R_+. \end{aligned}$$

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,

$$\begin{aligned} u_1(t)\le u(t),\ t\in \mathbb R_+. \end{aligned}$$

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

$$\begin{aligned} \rho _2(u,t,1)\le \max _{|t-\tau |\le \frac{1}{2}}\rho _2(u_1,\tau ,1)+2,\ t\in \mathbb R_+,\\ \rho _2(u_1,t,1)\le \max _{|t-\tau |\le \frac{1}{2}}\rho _2(u,\tau ,1)+2,\ t\in \mathbb R_+. \end{aligned}$$

Proof

Let

$$\begin{aligned} \max _{|t-\tau |\le \frac{1}{2}}\rho _2(u_1,\tau ,1)=M. \end{aligned}$$

Let us suppose that for a natural number n, satisfying $|n-t|\le \frac{1}{2}$, the following inequality holds

$$\begin{aligned} \rho _2(u,n,1)> [M]+1. \end{aligned}$$

Then, setting $k=[M]+1$, we have

$$\begin{aligned} \int \limits _{n-k}^{n+k}|u_+'(t )-u_+'(n)|dt<1, \end{aligned}$$

that is

$$\begin{aligned} u(n+k)+u(n-k)-2u(n)<1. \end{aligned}$$

Since the functions u and $u_1$ coincide at integer points, then

$$\begin{aligned} u_1(n+k)+u_1(n-k)-2u_1(n)<1. \end{aligned}$$

Hence,

$$\begin{aligned} \int \limits _{n-k}^{n+k}|(u_1)_+'(t )-(u_1)_+'(n)|dt<1, \end{aligned}$$

and

$$\begin{aligned} \rho _2(u_1,n,1)\ge k=[M]+1> M. \end{aligned}$$

The resulting contradiction means that

$$\begin{aligned} \rho _2(u,n,1)\le [M]+1\le M+1, \ n\in \mathbb N. \end{aligned}$$

Since the function $\rho _2(u,t,1)$ satisfies the Lipschitz condition, we have

$$\begin{aligned} \rho _2(u,t,1)\le \rho _2(u,[t],1)+|t-[t]|\le M+2, \ t\in \mathbb R_+. \end{aligned}$$

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

$$\begin{aligned} \sqrt{\frac{1}{q\widetilde{\psi }''(t)}}\le \rho _2(\widetilde{\psi },t,1)\le \sqrt{\frac{q}{\widetilde{\psi }''(t)}}. \end{aligned}$$

Proof

Let $\rho _0=\frac{\gamma (t)}{\sqrt{\widetilde{\psi }''(t)}}$, then due to regularity $\widetilde{\psi }$

$$\begin{aligned} \frac{1}{q} \le \frac{\widetilde{\psi }''(t)}{\widetilde{\psi }''(x)}\le q \,\,\, \text {for} \,\,\, |t-x|\le \rho _0. \end{aligned}$$

Hence, by the mean value theorem for any x such that $|x-t|\le \rho _0$ we have

$$\begin{aligned} |\widetilde{\psi }'(t)-\widetilde{\psi }'(x)|=\widetilde{\psi }''(x^*)|t-x|\ge \frac{1}{q}\widetilde{\psi }''(t)|t-x|. \end{aligned}$$

Therefore, if $\gamma (t)\ge q$, then

$$\begin{aligned} \int \limits _{t-\rho _0 }^{t+\rho _0}|\widetilde{\psi }'(x)-\widetilde{\psi }'(t)|dx\ge \frac{1}{q}\widetilde{\psi }''(t)\rho _0^2=\frac{1}{q}\gamma ^2(t)>1. \end{aligned}$$

Hence, $\rho _2(\widetilde{\psi },t,1):=\rho \le \rho _0$. By definition of the function $\rho _2(\widetilde{\psi },t,1)$ we have

$$\begin{aligned} \int \limits _{t-\rho }^{t+\rho }|\widetilde{\psi }'(x)-\widetilde{\psi }'(t)|dx=1, \end{aligned}$$

and

$$\begin{aligned} \left( \max _{|t-x|\le \rho }\widetilde{\psi }''(x)\right) ^{-1}\le \rho ^2\le \left( \min _{|t-x|\le \rho }\widetilde{\psi }''(x)\right) ^{-1}. \end{aligned}$$

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

$$\begin{aligned} \frac{1}{m}\rho _2(\widetilde{\psi },t+1,1)\le \rho _2(u_1 ,t,1) \le m \rho _2(\widetilde{\psi },t+1,1), \ t\in \mathbb R_+. \end{aligned}$$

The left estimate holds without the regularity condition.

Proof

1. Let us prove the left inequality. By Theorem 2(a) in [7] we have

$$\begin{aligned} e^{2u_1(y)}=\int \limits _{-\infty }^{\infty }e^{2(y+1)x-2\psi (x)}dx\asymp \frac{e^{2\widetilde{\psi }(y+1)}}{\rho _1(\widetilde{\psi },y+1,1)}, \ y\in \mathbb R_+. \end{aligned}$$

That is,

$$\begin{aligned} e^{-2a}\le \frac{e^{2(\widetilde{\psi }(y+1)-u_1(y))}}{\rho _1(\widetilde{\psi },y+1,1)}\le e^{2a}, \ y\in \mathbb R_+, \end{aligned}$$

for some $a>0$, and

$$\begin{aligned} \left| \widetilde{\psi }(y+1)-u_1(y)-\frac{1}{2} \ln \rho _1 (\widetilde{\psi },y+1,1)\right| \le a,\ y\in \mathbb R_+. \end{aligned}$$

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

$$\begin{aligned} \max _{x\in [t+1-\alpha \rho _1; t+1+\alpha \rho _1]}|\widetilde{\psi }(x)-l(x)|\le 1. \end{aligned}$$

For the linear function $l_1(x)=l(x)-\frac{1}{2}\ln \rho _1$ we have

$$\begin{aligned}&\max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}|u_1 (x)-l_1(x+1)|\\&\quad \le \max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}\left| u_1(x)-\widetilde{\psi }(x+1)+\frac{1}{2} \ln \rho _1(\widetilde{\psi },x+1,1)\right| \\&\quad +\max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}\left| \widetilde{\psi }(x+1)-l(x+1)\right| +\frac{1}{2}\max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}\left| \ln \frac{\rho _1(\widetilde{\psi },t+1,1)}{\rho _1 (\widetilde{\psi },x+1,1)}\right| \end{aligned}$$

$$\begin{aligned} \le a+1+\frac{1}{2}\max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}\left| \ln \frac{\rho _1(\widetilde{\psi },x+1,1)}{\rho _1(\widetilde{\psi },t+1,1)}\right| . \end{aligned}$$

(4)

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

$$\begin{aligned} |\rho _1 (\widetilde{\psi },x+1,1)-\rho _1(\widetilde{\psi },t+1,1)|\le \alpha \rho _1(\widetilde{\psi },t+1,1) \end{aligned}$$

or

$$\begin{aligned} \left| \frac{\rho _1(\widetilde{\psi },x+1,1)}{\rho _1(\widetilde{\psi },t+1,1)}-1\right| \le \alpha <\frac{1}{2}. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{2}\max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}\left| \ln \frac{\rho _1(\widetilde{\psi },x+1,1)}{\rho _1(\widetilde{\psi },t+1,1)}\right| \le \frac{1}{2}\max _{|s|<\frac{1}{2}}\left| \ln (1+s)\right| \le \frac{1}{2} \ln 2<\frac{1}{2}. \end{aligned}$$

Continuing estimate (4), we obtain

$$\begin{aligned} \max _{x\in [t-\alpha \rho _1; t+\alpha \rho _1]}|u_1 (x)-l_1(x+1)|<a+\frac{3}{2}, \end{aligned}$$

and by that,

$$\begin{aligned} \rho _1\left( u_1,t,a+\frac{3}{2}\right) \ge \alpha \rho _1(\widetilde{\psi },t+1,1),\ t\in \mathbb R_+, \end{aligned}$$

or taking into account the arbitrariness of $\alpha \in \left( 0;\frac{1}{2}\right)$, we get

$$\begin{aligned} \rho _1\left( u_1,t,a+\frac{3}{2}\right) \ge \frac{1}{2}\rho _1(\widetilde{\psi },t+1,1),\ t\in \mathbb R_+. \end{aligned}$$

By (1) we get

$$\begin{aligned} \rho _2(u_1,t,2a+3)\ge \frac{1}{2}\rho _2(\widetilde{\psi },t+1,2),\ t\in \mathbb R_+. \end{aligned}$$

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

$$\begin{aligned} \frac{1}{q}\le \frac{\widetilde{\psi }''(x+1)}{\widetilde{\psi }''(y+1)}\le q, \ |x-y|\le \gamma _1(x)\sqrt{\frac{1}{\widetilde{\psi }''(x+1)}}, \end{aligned}$$

where $\gamma _1(x)=\gamma (x+1)$. By Theorem 2(a) in [7] we have

$$\begin{aligned} e^{2u_1(y)}=\int \limits _{-\infty }^{\infty }e^{2(y+1)x-2\psi (x)}dx\asymp \frac{e^{2\widetilde{\psi }(y+1)}}{\rho _2 (\widetilde{\psi },y+1,1)}, \ y\in \mathbb R_+, \end{aligned}$$

that is, for some $b>0$ we have

$$\begin{aligned} e^{-2b}\le \frac{e^{2(\widetilde{\psi }(y+1)-u_1(y))}}{\rho _2(\widetilde{\psi },y+1,1)}\le e^{2b}, \ y\in \mathbb R_+, \end{aligned}$$

(5)

or

$$\begin{aligned} \left| \widetilde{\psi }(y+1)-u_1(y)-\frac{1}{2} \ln \rho _2(\widetilde{\psi },y+1,1)\right| \le b,\ y\in \mathbb R_+. \end{aligned}$$

By Lemma 4 we have

$$\begin{aligned} \frac{1}{q}\sqrt{\frac{\widetilde{\psi }''(y+1)}{\widetilde{\psi }''(t+1)} }\le \frac{\rho _2(\widetilde{\psi },t+1,1)}{\rho _2(\widetilde{\psi },y+1,1)}\le q\sqrt{\frac{\widetilde{\psi }''(y+1)}{\widetilde{\psi }''(t+1)} }, \end{aligned}$$

and by the regularity of $\widetilde{\psi }$ we get

$$\begin{aligned} q^{-\frac{3}{2} }\le \frac{\rho _2(\widetilde{\psi },t+1,1)}{\rho _2(\widetilde{\psi },y+1,1)}\le q^{\frac{3}{2} }, \ \ |t-y|\le \gamma _1(t)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}. \end{aligned}$$

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

$$\begin{aligned} e^{-2(b+c)}\le e^{2(\widetilde{\psi }(y+1)-\frac{1}{2}\ln \rho _2-u_1(y))}\le e^{2(b+c)}, \ \ |t-y|\le \gamma _1(t)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}, \end{aligned}$$

or

$$\begin{aligned} \left| \widetilde{\psi }(y+1)-\frac{1}{2}\ln \rho _2 -u_1(y)\right| \le b+c,\ \ |t-y|\le \gamma _1(t)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}. \end{aligned}$$

(6)

Suppose that

$$\begin{aligned} \rho _1(u_1,t,1)\ge 3 q(b+c+1)\rho _2(\widetilde{\psi }, t+1,1). \end{aligned}$$

Then there is a linear function l(x) such that

$$\begin{aligned} |u_1(x)-l(x)|\le 1,\ |x-t|\le 3 q(b+c+1)\rho _2(\widetilde{\psi }, t+1,1). \end{aligned}$$

By Lemma 4 we have

$$\begin{aligned} |u_1(x)-l(x)|\le 1,\ |x-t|\le 3\sqrt{q}(b+c+1)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}. \end{aligned}$$

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

$$\begin{aligned}&|\widetilde{\psi }(x+1)-l_1(x)|= \left| \widetilde{\psi }(x+1)-\frac{1}{2}\ln \rho _2 -l(x)\right| \\&\quad \le \left| \widetilde{\psi }(x+1)-\frac{1}{2}\ln \rho _2-u_1(x)\right| +|u_1(x)-l(x)|\le b+c+1 \end{aligned}$$

for $|x-t|\le 3\sqrt{q}(b+c+1)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}$. Hence,

$$\begin{aligned} \rho _1(\widetilde{\psi },t+1,b+c+1)\ge 3\sqrt{q}(b+c+1)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}, \end{aligned}$$

and by (1),

$$\begin{aligned} 3\sqrt{q}(b+c+1)\sqrt{\frac{1}{\widetilde{\psi }''(t+1)}}\le \rho _2 (\widetilde{\psi },t+1,2(b+c+1)). \end{aligned}$$

Then by Lemma 4 we get

$$\begin{aligned} 3(b+c+1)\rho _2 (\widetilde{\psi },t+1,1)\le \rho _2 (\widetilde{\psi },t+1,2(b+c+1)). \end{aligned}$$

Hence, by Lemma 2 in [7] we obtain

$$\begin{aligned} 3(b+c+1)\rho _2(\widetilde{\psi }, t+1,1)\le \rho _2(\widetilde{\psi },t+1,2(b+c+1))\le 2(b+c+1)\rho _2(\widetilde{\psi }, t+1,1) \end{aligned}$$

or

$$\begin{aligned} \frac{3}{2}\rho _2(\widetilde{\psi }, t+1,1)\le \rho _2(\widetilde{\psi }, t+1,1). \end{aligned}$$

Since $\rho _2(\widetilde{\psi }, t+1,1)>0$, we obtain a contradiction. Thus,

$$\begin{aligned} \rho _1(u_1,t,1)< 3q(b+c+1)\rho _2(\widetilde{\psi }, t+1,1). \end{aligned}$$

Taking into account (1) again, for t such that $\gamma _1(t)\ge 3\sqrt{q}(b+c+1)$ we have

$$\begin{aligned} \rho _2(u_1,t,1)\le \rho _2(u_1,t,2)=\rho _1(u_1,t,1)< 3\sqrt{q}(b+c+1)\rho _2(\widetilde{\psi }, t+1,1). \end{aligned}$$

Since the functions $\rho _2(u_1,t,1)$ and $\rho _2(\widetilde{\psi }, t,1)$ are continuous, this implies the estimate

$$\begin{aligned} \rho _2(u_1,t,1)\le A\rho _2(\widetilde{\psi }, t+1,1), \ t\in \mathbb R_+, \end{aligned}$$

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

$$\begin{aligned} \inf _{t>0}\widetilde{\psi }''(t)>0, \end{aligned}$$

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

$$\begin{aligned} \sup _{t>0} \psi ''(t)<\infty , \end{aligned}$$

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

$$\begin{aligned} v'(\widetilde{v}'(t))\equiv t,\ \ \quad v''(\widetilde{v}'(t))\widetilde{v}''(t)\equiv 1,\ t\in \mathbb R. \end{aligned}$$

(7)

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

$$\begin{aligned} \lim _{x\rightarrow +\infty }t_+(x)=+\infty . \end{aligned}$$

(8)

By regularity of v, for some $x^*\in [x_-;x_+]$ we have

$$\begin{aligned} t_+-t_-=v''(x^*)(x_+-x_-)\ge \frac{\gamma (x)}{q}\sqrt{v''(x)}. \end{aligned}$$

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

$$\begin{aligned} \lim _{x\rightarrow +\infty }\tau (x)=+\infty . \end{aligned}$$

(9)

If

$$\begin{aligned} |\tau -s|\le \frac{\gamma (x)}{2q^{\frac{3}{2}}\sqrt{\widetilde{v}''(\tau )}}, \end{aligned}$$

then by (7) we get

$$\begin{aligned} |\tau -s|\le \frac{\gamma (x)}{2q^{\frac{3}{2}}}\sqrt{v''(y)}\le \frac{\gamma (x)}{2q}\sqrt{v''(x)}, \end{aligned}$$

that is $s\in [t_-;t_+]$ and $\widetilde{v}'(s)\in [x_-;x_+]$. Hence, by (7) we obtain

$$\begin{aligned} \frac{\widetilde{v}''(\tau )}{\widetilde{v}''(s )}=\frac{ v''(\widetilde{v}'(s))}{v''(\widetilde{v}'(\tau ))}\in \left[ \frac{1}{q};q\right] . \end{aligned}$$

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

$$\begin{aligned} \gamma _0(\tau )=\frac{1}{2q^{\frac{3}{2}}}\inf _{t\ge \tau } \gamma (x(t)). \end{aligned}$$

$\square$

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

$$\begin{aligned} 0<\psi ''(t)\longrightarrow 0, \ t\longrightarrow \infty ,\\ |\psi '''(t)|=O(\psi ''(t)),\ t\longrightarrow \infty , \end{aligned}$$

then the Fock space with the weight $\psi (\ln |\lambda |)$ possesses Riesz bases of normalized reproducing kernels.

Proof

By (7) we obtain that

$$\begin{aligned} \widetilde{\psi }''(x)\longrightarrow \infty ,\ x\longrightarrow \infty , \end{aligned}$$

and for some $M>0$ we have

$$\begin{aligned} |(\ln \widetilde{\psi }''(\psi '(t)))'|=\left| \frac{\psi '''(t)}{\psi ''(t)}\right| \le M, \ t\in \mathbb R. \end{aligned}$$

(10)

Hence, by mean value theorem we have

$$\begin{aligned} \left| \ln \left( \frac{\widetilde{\psi }''(x)}{\widetilde{\psi }''(y)}\right) \right| \le M|x-y|,\ x,y \in \mathbb R_+. \end{aligned}$$

Let

$$\begin{aligned} \gamma (x)=\inf _{y\ge x}\ln \widetilde{\psi }''(y), \ x>0. \end{aligned}$$

Then $\gamma (x)\uparrow +\infty$ as $x \rightarrow +\infty$, and $\gamma (x)\le \ln \widetilde{\psi }''(x)$. Put

$$\begin{aligned} C=\sup _{x>0}\frac{\gamma (x)}{\sqrt{\widetilde{\psi }''(x)}}, \end{aligned}$$

then $C<\infty$. If

$$\begin{aligned} |x-y|\le \frac{\gamma (x)}{\sqrt{\widetilde{\psi }''(x)}},\, x,y\in \mathbb R_+, \end{aligned}$$

then $|x-y|\le C$. Hence, by (10) we have

$$\begin{aligned} \left| \ln \left( \frac{\widetilde{\psi }''(x)}{\widetilde{\psi }''(y)}\right) \right| \le M|x-y|\le MC. \end{aligned}$$

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

$$\begin{aligned} |\psi '''(t)|=O\left( \psi ''(t)^{\frac{5}{3}}\right) , \ t\longrightarrow \infty . \end{aligned}$$

Monotonicity implies the existence of a limit

$$\begin{aligned} \lim _{t\longrightarrow \infty }\psi ''(t):=\psi _0. \end{aligned}$$

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 $\psi '''$.