Necessary Density Conditions for Sampling and Interpolation of Certain Entire Functions with Doubling Weights

Abstract

Let $\Lambda \subseteq {\mathbb{R}}^d$ be a discrete uniformly separated subset. In the unweighted case and $p=2$, Landau obtained the necessary conditions for sampling and interpolation of functions in Paley–Wiener space in terms of the upper and lower uniform densities of $\Lambda$. In this paper, we generalize the above results to the weighted case, and give some necessary density conditions for $L_p$ weighted sampling and interpolating sets for all $0<p<\infty$.

1. Introduction

Let w be a weight on ${\mathbb{R}}^d$. For $0<p<\infty ,$ we denote by $L_{p,w}\equiv L_{p,w}\left({\mathbb{R}}^d\right)$ the space of all Lebesgue measurable functions on ${\mathbb{R}}^d$ with finite quasi-norms

$${\parallel f\parallel }_{p,w}={\left({\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{p}w\left(x\right)dx\right)}^{1/p}.$$

For the unweighted case, i.e., $w\left(x\right)\equiv 1$, we write $L_p$ and ${\parallel ·\parallel }_p$ instead of $L_{p,w}$ and ${\parallel ·\parallel }_{p,w}$ for simplicity.

Let $\mathcal{S}=\mathcal{S}\left({\mathbb{R}}^d\right)$ be the Schwartz space of all complex-valued rapidly decreasing functions in ${\mathbb{R}}^d$, and let ${\mathcal{S}}^{\prime }={\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ be the complex-valued tempered distributions. For $\sigma >0$ and $0<p<\infty ,$ the weighted Paley–Wiener space $P{W}_{p,w}^{\sigma }\equiv P{W}_{p,w}^{\sigma }\left({\mathbb{R}}^d\right)$ consisting of all entire functions of spherical exponential type $\sigma$ whose restrictions to ${\mathbb{R}}^d$ are in $L_{p,w}$ is defined by

$$P{W}_{p,w}^{\sigma }:=\left\{f\in {\mathcal{S}}^{\prime }|\mathrm{supp}\mathcal{F}f\subset B(0,\sigma ),{\parallel f\parallel }_{p,w}<\infty \right\},$$

where $\mathcal{F}f$ is the Fourier transform of the distribution $f\in {\mathcal{S}}^{\prime }$, and $B(x,r)=\{y\in {\mathbb{R}}^d||x-y|\le r\}$ (with Euclidean distance $|x-y|$ on ${\mathbb{R}}^d$) is the (closed) ball with center x and radius r. We simply write $P{W}_p^{\sigma }$ if $w\left(x\right)\equiv 1$.

Let $\Lambda$ be a discrete subset of ${\mathbb{R}}^d$. We say that $\Lambda$ is an $L_p$-sampling set for $P{W}_p^{\sigma }$, if there exists a constant $C>0$ such that for all $f\in P{W}_p^{\sigma }$,

$${\displaystyle \frac{1}{C}}\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p\le {\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^p)dx\le C\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p.$$

We say that $\Lambda$ is an $L_p$-interpolating set for $P{W}_p^{\sigma }$, if there exists a constant $C>0$ such that for each sequence ${\left\{c_{\xi }\right\}}_{\xi \in \Lambda }$ satisfying ${\sum }_{\xi \in \Lambda }{\left|c_{\xi }\right|}^p<\infty ,$ there is a function $f\in P{W}_p^{\sigma }$, for which $f\left(\xi \right)=c_{\xi },\xi \in \Lambda$ and

$${\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{p}dx\le C\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p.$$

Intuitively, a sampling set should be dense in order to recover the $L_p$-norm of functions of the space $P{W}_p^{\sigma }$, and an interpolating set should be sparse. The $L_p$-sampling condition is also known as Marcinkiewicz–Zygmund inequalities for the trigonometric polynomials, for spaces of algebraic polynomials with respect to some measure, and for spaces of spherical harmonics on the sphere. The existence of a $L_p$-sampling set, $\Lambda$ means that the $L_p$-norm of $P{W}_p^{\sigma }$ is comparable to the discrete version given by the $l_p$-norm of its restriction to $\Lambda$. It follows from Theorem 2 (see Section 2) that such a $L_p$-sampling set for $P{W}_p^{\sigma }$ exists if the set is dense enough. A good reference for material on sampling and interpolation in spaces of analytic functions is [1].

The sampling and interpolation in the Paley–Wiener setting provide a mathematical model of stable recovery and data transmission in signal theory regardless of the computation method. In 1967, Landau’s classical results ([2]) showed the necessary conditions for the sampling and interpolation of certain entire functions of Paley–Wiener space on Euclidean space ${\mathbb{R}}^d$ in terms of uniform densities for the unweighted case $w\left(x\right)=1$ with $p=2$, which established Beurling’s conjecture regarding the density of sets of balayage for greater dimensions, also known as the precise mathematical formulation of the Nyquist density ([3]). While a great deal of work has extended the theory of Beurling–Landau on the discretization of functions in the Paley–Wiener space on ${\mathbb{R}}^d$ to functions in many contexts (see [4,5,6,7,8,9,10,11,12,13] and references therein).

However, these foundational results of Landau are limited to the unweighted $L_2$ setting. Many real-world applications require the analysis of non-uniformly distributed data or signals with spatially varying energy. This naturally leads to weighted function spaces, where the weight quantifies the local importance or measurement density. For instance, in sensor networks with irregular node distributions, weights model spatial sampling density. In signal processing, weights arise in non-uniform sampling reconstruction and spectral estimation with prior information. In approximation theory, doubling weights (defined later) cover critical cases like polynomial weights, which are essential for handling singularities or boundary effects.

Despite this practical relevance, there are few results concerning the sampling and interpolation conditions on weighted spaces, especially for the weighted Paley–Wiener spaces on ${\mathbb{R}}^d$, for general $0<p<\infty$ and doubling weights. Recent work by Dai and Wang ([14]) addressed the interpolation for weighted Paley–Wiener spaces associated with the Dunkl transform in one dimension, and Wang ([15]) provided the necessary density conditions on the sphere. Motivated by these results and applications, in this paper, we mainly discuss the sampling and interpolation of certain entire functions with doubling weights on ${\mathbb{R}}^d$. The problem addressed in this paper is the lack of necessary density conditions for sampling and interpolation sets in the general weighted Paley–Wiener space $P{W}_{p,w}^{\Omega }$ on ${\mathbb{R}}^d$ for all $0<p<\infty$ and doubling weights w (for the definition of $P{W}_{p,w}^{\Omega }$ and w, see Section 2). We aim to establish such necessary conditions, generalizing Landau’s fundamental results to this broader setting. Generalizing Landau’s density theory to weighted settings is fundamental for two reasons: (i) It establishes whether density constraints are intrinsic to the spectral domain $\Omega$ or altered by the weight w. Our results show universality—density thresholds depend only on $\Omega$, independent of w, which means that the weighted and unweighted cases share the same density properties. (ii) Necessary density conditions guide optimal sensor placement in weighted sensing frameworks and certify minimal sampling rates for reconstruction guarantees in $L_{p,w}$-norms.

The outline of this paper is as follows. In the next section, we recall some preliminary knowledge and summarize some known facts about the special maximal function. The Plancherel–Pólya-type inequality with doubling weight and its corollary are also introduced for the separation of the sampling set. We give our main result (Theorem 3) and its proof in Section 3 where, by enlarging or slightly diminishing the measurable set $\Omega$, we obtain an $L_{q,\tilde{w}}$-sampling or $L_{q,\tilde{w}}$-interpolating sets starting from any other $L_{p,w}$-sampling or $L_{p,w}$-interpolating sets for $P{W}_{p,w}^{\Omega }$. Then, the proof of Theorem 3 can be finished by the lemmas in Section 3 and Landau’s results. Finally, a conclusion in Section 4 provides a summary of the key findings, highlights the contributions of the work, and suggests potential directions for future research.

Throughout the paper, assume that C denotes a constant which may be different in different occurrences even in the same line, and $A\sim B$ means that there exists a constant $C>0$ such that ${\displaystyle \frac{1}{C}}A\le B\le CA$, where C is called the constant of equivalence.

2. Preliminaries

In this section, we recall some basic knowledge and lemmas. First, we give some notations. For a measurable subset E of ${\mathbb{R}}^d$, let $\left|E\right|$ denote the Lebesgue measure of E, and $\#E$ denote the number of elements in E. Write $w\left(E\right)$ the weighted measure of E, i.e.,

$$w\left(E\right)={\int }_{E}w\left(t\right)dt.$$

2.1. Doubling Weights

We call a weight $w\left(x\right)$ on ${\mathbb{R}}^d$ a doubling weight if there exists a constant $L\ge 0$ such that, for any $x\in {\mathbb{R}}^d$ and any $r>0$, we have

$$w\left(B\right(x,2r\left)\right)\le Lw\left(B\right(x,r\left)\right),$$

where the least constant $L=L_w$ is called the doubling constant of w.

In the remainder of this paper, we always assume $\sigma >0$, and w to be a doubling weight. For any $x\in {\mathbb{R}}^d$, define

$$w_{\sigma }\left(x\right)={\displaystyle \frac{1}{\left|B\right(x,1/\sigma \left)\right|}}{\int }_{B(x,{\displaystyle \frac{1}{\sigma }})}w\left(t\right)dt.$$

(1)

For arbitrary $x,y\in {\mathbb{R}}^d$ and $\sigma >0$, choose an integer m such that $2^m\le 1+\sigma |x-y|\le 2^{m+1}$. The doubling condition for w implies that

$$\begin{array}{cc}\hfill w(B(x,{\displaystyle \frac{1}{\sigma }}))& \le w(B(y,|x-y|+{\displaystyle \frac{1}{\sigma }}))\le w(B(y,{\displaystyle \frac{2^{m+1}}{\sigma }}))\hfill \\ & \le L^{m+1}w(B(y,{\displaystyle \frac{1}{\sigma }}))\le L{(1+\sigma |x-y|)}^{logL/log2}w(B(y,{\displaystyle \frac{1}{\sigma }})),\hfill \end{array}$$

which means that

$$w_{\sigma }\left(x\right)\le L{(1+\sigma |x-y\left|\right)}^s{w}_{\sigma }\left(y\right),$$

(2)

with $s=logL/log2$. We note that each $w_{\sigma }(·)$ is again a doubling weight with the doubling constant depending only on d and that of w.

Denote by $E_{\sigma }$ the space of all slowly increasing entire functions of spherical exponential type $\sigma$, that is, $E_{\sigma }=\{f\in {\mathcal{S}}^{\prime }|\mathrm{supp}\mathcal{F}f\subset B(0,\sigma )\}$. Now, we introduce another lemma for which $w_{\sigma }$ can be replaced by the pth power of the absolute value of an entire function $g\left(x\right)$ of spherical exponential type $\sigma$.

  Lemma 1 

([16] (Lemma 2.5)). Let $\sigma >0$. Let α be a fixed non-negative number and f be a non-negative function on ${\mathbb{R}}^d$ satisfying

$$f\left(x\right)\le C_1{(1+\sigma |x-y\left|\right)}^{\alpha }f\left(y\right),x,y\in {\mathbb{R}}^d.$$

Then, for any $0<p<\infty$, there exists a function $g\in E_{\sigma }$ such that

$$C^{-1}f\left(x\right)\le {\left(g\left(x\right)\right)}^p\le Cf\left(x\right),x\in {\mathbb{R}}^d,$$

where $C>0$ depends only on d, $C_1$, p, and α. Specifically, for a doubling weight w and any $0<p<\infty$, there exists a function $g_w\in E_{\sigma }$ such that

$$C^{-1}{w}_{\sigma }\left(x\right)\le {\left(g_w\left(x\right)\right)}^p\le C{w}_{\sigma }\left(x\right),x\in {\mathbb{R}}^d,$$

(3)

where $C>0$ depends only on d, p, and the doubling constant L.

Remark 1. 

For the one-dimensional case, it is easily seen that, for fixed $y\in \mathbb{R},$ there exists an entire function $g\in E_{\sigma }$ such that

$$C^{-1}{(1+\sigma |x-y\left|\right)}^{\alpha }\le g(x-y)\le C{(1+\sigma |x-y\left|\right)}^{\alpha }.$$

2.2. Maximal Function

In this part, we introduce a special maximal function on ${\mathbb{R}}^d$, which has been used in the inequality problems of the weighted Paley–Wiener space on ${\mathbb{R}}^d$ (see [16]). In fact, the maximal function defined on the sphere and on the ball has been used in the density conditions derivation problems of Marcinkiewicz–Zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights (see [15,17]).

For $\beta >0$, $f\in C\left({\mathbb{R}}^d\right)$, and s being the constant in (2), the maximal function $f_{\beta ,\sigma }^{*}\left(x\right)$ is defined by

$$f_{\beta ,\sigma }^{*}\left(x\right)=\underset{z\in {\mathbb{R}}^d}{max}{\displaystyle \frac{\left|f\right(z\left)\right|}{{(1+\sigma |x-z\left|\right)}^{\beta s}}}.$$

It is easy to verify that, for any $x,y\in {\mathbb{R}}^d$,

$$f_{\beta ,\sigma }^{*}\left(x\right)\le {(1+\sigma |x-y\left|\right)}^{\beta s}{f}_{\beta ,\sigma }^{*}\left(y\right).$$

(4)

For $\xi \in {\mathbb{R}}^d$, when $x\in B(\xi ,1/\sigma )$, it follows that

$$f_{\beta ,\sigma }^{*}\left(x\right)\sim f_{\beta ,\sigma }^{*}\left(\xi \right).$$

(5)

The following theorem shows that the weighted norm of $f\in P{W}_{p,w}^{\sigma }$ and the corresponding maximal functions $f_{\beta ,\sigma }^{*}$ with respect to the weight w and $w_{\sigma }$ are comparable. This result was first proven in [16].

Theorem 1 

([16] (Theorems 2.1 and 2.2)). For $0<p<\infty ,f\in P{W}_{p,w}^{\sigma }$ and $\beta >1/p$, we have

$${\parallel f\parallel }_{p,w}\sim \parallel f_{\beta ,\sigma }^{*}{\parallel }_{p,w}\sim {\parallel f\parallel }_{p,w_{\sigma }}\sim {\parallel f_{\beta ,\sigma }^{*}\parallel }_{p,w_{\sigma }},$$

(6)

where the constants of equivalence depend only on $d,p$, the doubling constant L and β when β is close to $1/p$.

To measure the sparsity of the uniform separation between points of the same generation, we introduce the following definition.

Given $\epsilon >0$, we say a discrete $\Lambda \subset {\mathbb{R}}^d$ is (uniformly) $\epsilon$-separated if for any $\xi ,{\xi }^{\prime }\in \Lambda ,\xi \ne {\xi }^{\prime },|\xi -{\xi }^{\prime }|>\epsilon .$

We say that a discrete $\Lambda \subset {\mathbb{R}}^d$ is maximal $\epsilon$-separated if it is $\epsilon$-separated and satisfies

$$\underset{x\in {\mathbb{R}}^d}{max}\underset{\xi \in \Lambda }{min}|\xi -x|\le \epsilon .$$

Obviously, for a maximal $\epsilon$-separated subset $\Lambda$ of ${\mathbb{R}}^d$, we have

$$1\le {\Lambda }_{\epsilon }\left(x\right):=\sum _{\xi \in \Lambda }{\chi }_{B(\xi ,\epsilon )}\left(x\right)\le C_d,\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^d.$$

We recall some results of the Plancherel–Pólya-type inequality with doubling weight.

Theorem 2 

([16] (Theorem 4.1)). Let $0<p<\infty$. Then, there exists a constant ${\delta }_0>0$ depending only on d and the doubling constant of w such that for any $\delta \in (0,{\delta }_0)$, any maximal ${\displaystyle \frac{\delta }{\sigma }}$-separated subset Λ of ${\mathbb{R}}^d$, and any ${\xi }_{\omega }\in B(\omega ,{\displaystyle \frac{\delta }{\sigma }})$,

$${\parallel f\parallel }_{p,w}^p\sim \sum _{\omega \in \Lambda }{\left|f\left({\xi }_{\omega }\right)\right|}^p{\int }_{B(\omega ,{\displaystyle \frac{1}{\sigma }})}w\left(x\right)dx$$

(7)

holds for all $f\in P{W}_{p,w}^{\sigma }$, where the constants of equivalence depend only on $d,p,\delta ,$ and the doubling constant of w.

Corollary 1 

([16] (Corollary 4.5)). Let Λ be a finite subset of ${\mathbb{R}}^d$ satisfying (7). Then, there exists a $\delta /\sigma$-separated subset $\tilde{\Lambda }$ of Λ which also satisfies (7) for some $\delta >0$.

Let $\eta$ be a non-negative $C^{\infty }$-function on ${\mathbb{R}}^d$ with the properties of $\mathcal{F}\eta \left(x\right)=1$ for $\left|x\right|\le 1$ and $\mathcal{F}\eta \left(x\right)=0$ for $x\ge 2$. Define for $\sigma >0$, $K_{\sigma }\left(x\right)={\sigma }^d\eta \left(\sigma x\right).$ For $f\in P{W}_{p,w}^{\sigma }$, by Palay–Wiener–Schwartz theorem (see [18]), there exist constants $\lambda ,c>0$, such that for any $x\in {\mathbb{R}}^d$,

$$f\left(x\right)\le c{(1+|x\left|\right)}^{\lambda }.$$

Since $K_{\sigma }$ is rapidly decreasing, we find that the integral ${\int }_{{\mathbb{R}}^d}f\left(t\right)K_{\sigma }(x-t)dt$ makes sense and is also slowly increasing. Furthermore, we have

$$\mathcal{F}f=\mathcal{F}f·\mathcal{F}{K}_{\sigma },$$

which means that

$$f\left(x\right)={\int }_{{\mathbb{R}}^d}f\left(t\right)K_{\sigma }(x-t)dt,x\in {\mathbb{R}}^d.$$

(8)

For any positive integer l and $x\in {\mathbb{R}}^d$, we have

$$|\nabla K_{\sigma }\left(x\right)|={\sigma }^{d+1}|\nabla \eta \left(\sigma x\right)|\le C{\sigma }^{d+1}{(1+|\sigma x\left|\right)}^{-l}.$$

For $y,u\in {\mathbb{R}}^d$, if $z\in B(y,1/\sigma )$, then

$$\begin{array}{cc}\hfill |K_{\sigma }(y-u)-K_{\sigma }(z-u)|& \le \underset{t\in (0,1)}{sup}C{\sigma }^{d+1}{(1+\sigma |(y-u)+(1-t)(z-y)\left|\right)}^{-l}\hfill \\ \hfill & \le C{\sigma }^{d+1}{|y-z|(1+\sigma |y-u\left|\right)}^{-l}.\hfill \end{array}$$

(9)

2.3. Weighted $L_{p,w}$-Sampling and $L_{p,w}$-Interpolating Sets

Let $\Omega$ be a measurable bounded subset of ${\mathbb{R}}^d$ with $|\Omega |>0$ and $|\mathsf{\partial }\Omega |=0$, where $\mathsf{\partial }\Omega$ is the boundary of $\Omega$. The weighted Paley–Wiener space $P{W}_{p,w}^{\Omega }\equiv P{W}_{p,w}^{\Omega }\left({\mathbb{R}}^d\right)$ is defined by

$$P{W}_{p,w}^{\Omega }:=\left\{f\in {\mathcal{S}}^{\prime }|\mathrm{supp}\mathcal{F}f\subset \Omega ,{\parallel f\parallel }_{p,w}<\infty \right\}.$$

Let $\Lambda$ be a discrete subset of ${\mathbb{R}}^d$. We say that $\Lambda$ is an $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$, if there exists a constant $C>0$ such that for all $f\in P{W}_{p,w}^{\Omega }$,

$${\displaystyle \frac{1}{C}}\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{w}_1\left(\xi \right)\le {\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{p}w\left(x\right)dx\le C\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{w}_1\left(\xi \right),$$

(10)

where

$$w_1\left(\xi \right)={\displaystyle \frac{1}{\left|B\right(\xi ,1\left)\right|}}{\int }_{B(\xi ,1)}w\left(t\right)dt.$$

(11)

We say that $\Lambda$ is an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$, if there exists a constant $C>0$ such that for each sequence ${\left\{c_{\xi }\right\}}_{\xi \in \Lambda }$ satisfying

$$\sum _{\xi \in \Lambda }{\left|c_{\xi }\right|}^p{w}_1\left(\xi \right)<\infty ,$$

there is a function $f\in P{W}_{p,w}^{\Omega }$, for which $f\left(\xi \right)=c_{\xi },\xi \in \Lambda$ and

$${\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{p}w\left(x\right)dx\le C\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{w}_1\left(\xi \right).$$

(12)

We note that the above definitions for sampling and interpolation sets are identical with the classical ones when $p=2$ and $w\left(x\right)=1$ (see [2]). When the discrete set $\Lambda$ is both $L_{p,w}$-sampling and $L_{p,w}$-interpolating set, we say that $\Lambda$ is a complete $L_{p,w}$-interpolating set. In the one-dimensional and unweighted case, there is a complete characterization for $L_p$-complete interpolating sequences for one-dimensional Paley–Wiener spaces in [19,20].

3. Main Results

Following Landau’s definition, let $\Lambda \subset {\mathbb{R}}^d$ be a uniformly separated subset, we define the upper and lower Beurling uniform density, respectively, as

$$D^{+}(\Lambda )=\underset{h\to \infty }{lim\; sup}{\displaystyle \frac{\underset{x\in {\mathbb{R}}^d}{sup}\#(\Lambda \cap Q_h\left(x\right))}{h^d}};$$

and

$$D^{-}(\Lambda )=\underset{h\to \infty }{lim\; inf}{\displaystyle \frac{\underset{x\in {\mathbb{R}}^d}{inf}\#(\Lambda \cap Q_h\left(x\right))}{h^d}},$$

where $Q_h\left(x\right)$ is the cube centered at x of sidelength $h>0$.

In the unweighted case and $p=2$, Landau in [2] found the necessary conditions for sampling and interpolation for functions in $P{W}_2^{\Omega }$ in terms of the uniform upper and lower density of $\Lambda$. In this paper, we generalize the results of Landau to the weighted case and give some necessary density conditions for the $L_{p,w}$-sampling and interpolating sequences for all $0<p<\infty$. The weighted case is crucial for many practical applications where measurements or reconstructions are inherently non-uniform. Sampling and interpolation with weights allow for prioritizing certain regions of space, reflecting varying importance, measurement reliability, or underlying density. Establishing necessary density conditions for sampling and interpolation in the weighted setting is essential for extending the applicability of the Paley–Wiener framework to these and other scenarios involving non-uniform data or reconstruction priorities. Weighted sampling and interpolation formalize this focus, and the associated density theorems provide theoretical guarantees. Our main result can be formulated as follows.

Theorem 3. 

Let $0<p<\infty$, w be a doubling weight, and Λ be a discrete subset of ${\mathbb{R}}^d$.

(i) If Λ is an $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$, there exists a uniformly separated sampling subsequence ${\Lambda }^{\prime }\subset \Lambda$ such that

$$D^{-}\left({\Lambda }^{\prime }\right)\ge {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}}.$$

(ii) If Λ is an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$, then Λ is uniformly separated and

$$D^{+}(\Lambda )\le {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}}.$$

Our results for the densities of the sampling and interpolation of the weighted Paley–Wiener space show a sort of universality with the unweighted case, and hold for all $0<p<\infty$ and for doubling weights w, significantly extending Landau’s classical results which were confined to $p=2$ and $w=1$. The proof of Theorem 3 is heavily dependent on Landau’s results ([2]). However, we need to transform the results from the weighted case to the unweighted case. A special maximal function is necessary, as the proofs rely on establishing connections between the weighted norm and a specially constructed maximal function (Theorem 1), and leveraging the properties of doubling weights, which plays an important role in several key related lemmas. The techniques for the maximal function have been used in [21,22] for the problems of approximation theory and harmonic analysis on spheres and balls.

3.1. Perturbative Results About Interpolating and Sampling Sets

The proof of Theorem 3 follows directly from the classical results of Landau in [2], Corollary 1, and the three following lemmas.

Lemma 2. 

Let Λ be an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$. Then, Λ is uniformly ε-separated for some $\epsilon >0$.

Proof. 

We suppose that $\Omega \subset B(0,\sigma )$ for some positive constant $\sigma$. Since $\Lambda$ is an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$, by definition, we obtain that for each $\xi \in \Lambda$, there exists a function $f_{\xi }\in P{W}_{p,w}^{\Omega }\subset P{W}_{p,w}^{\sigma }$ such that

$$f_{\xi }\left({\xi }^{\prime }\right)={\delta }_{\xi ,{\xi }^{\prime }},{\xi }^{\prime }\in \Lambda ,$$

with

$$\parallel f_{\xi }{\parallel }_{p,w}^p\le c{w}_1\left(\xi \right)\le c{\int }_{B(\xi ,1/\sigma )}w\left(x\right)dx.$$

(13)

It suffices to show that there exists a constant ${\epsilon }^{\prime }>0$ such that

$$\begin{array}{c}\hfill |\xi -{\xi }^{\prime }|\ge {\epsilon }^{\prime }/\sigma ,\xi \ne {\xi }^{\prime },\end{array}$$

(14)

for all ${\xi }^{\prime }$ satisfying $|\xi -{\xi }^{\prime }|\le 1/\sigma$. By (5), (6), and (13), we obtain that

$$\begin{array}{cc}\hfill {\left({\left(f_{\xi }\right)}_{2/p,\sigma }^{*}\left(\xi \right)\right)}^p& \le c{\displaystyle \frac{{\int }_{B(\xi ,1/\sigma )}{\left({\left(f_{\xi }\right)}_{2/p,\sigma }^{*}\left(x\right)\right)}^{p}w\left(x\right)dx}{{\int }_{B(\xi ,1/\sigma )}w\left(x\right)dx}}\hfill \\ \hfill & \le c{\displaystyle \frac{\parallel {\left(f_{\xi }\right)}_{2/p,\sigma }^{*}{\parallel }_{p,w}^p}{{\int }_{B(\xi ,1/\sigma )}w\left(x\right)dx}}\le c{\displaystyle \frac{\parallel f_{\xi }{\parallel }_{p,w}^p}{{\int }_{B(\xi ,1/\sigma )}w\left(x\right)dx}}\le c.\hfill \end{array}$$

(15)

It follows from (8), (9) and (15) that for any ${\xi }^{\prime }\in B(\xi ,1/\sigma )$,

$$\begin{array}{cc}\hfill 1=|f_{\xi }\left(\xi \right)-f_{\xi }\left({\xi }^{\prime }\right)|& \le c|\xi -{\xi }^{\prime }|{\int }_{{\mathbb{R}}^d}|f_{\xi }\left(t\right)|{\sigma }^{d+1}{(1+\sigma |\xi -t\left|\right)}^{-l}dt\hfill \\ \hfill & \le c|\xi -{\xi }^{\prime }|{\left(f_{\xi }\right)}_{2/p,\sigma }^{*}\left(\xi \right){\int }_{{\mathbb{R}}^d}{\sigma }^{d+1}{(1+\sigma |\xi -t\left|\right)}^{-l+2s/p}dt\hfill \\ \hfill & \le c\sigma |\xi -{\xi }^{\prime }|{\left(f_{\xi }\right)}_{2/p,\sigma }^{*}\left(\xi \right)\le c\sigma |\xi -{\xi }^{\prime }|\hfill \end{array}$$

proving (14). The proof of Lemma 2 is complete. □

Let $0<p,q<\infty$, w and $\tilde{w}$ be two doubling weights and let $0<p,q<\infty$. We shall show that by slightly enlarging or diminishing the measurable set $\Omega$ we can get the $L_{q,\tilde{w}}$-sampling or $L_{q,\tilde{w}}$-interpolating sets starting from any other $L_{p,w}$-sampling or $L_{p,w}$-interpolating sets for $P{W}_{p,w}^{\Omega }$.

For $\epsilon >0$, let ${\Omega }_{\epsilon }=\left\{z\right|z=x+y,x\in \Omega ,y\in B(0,\epsilon )\}$ and ${\Omega }_{-\epsilon }=\{x\in \Omega |{inf}_{y\in \mathsf{\partial }\Omega }|x-y|\ge \epsilon \}$.

Lemma 3. 

Let $0<p,q<\infty$ , w and $\tilde{w}$ be two doubling weights, and let Λ be a $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$. Then for $\epsilon >0$, Λ is a $L_{q,\tilde{w}}$-interpolating set for $P{W}_{q,\tilde{w}}^{{\Omega }_{\epsilon }}$.

Proof. 

Since $\Lambda$ is an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$, by definition we get that for each $\xi \in \Lambda$, there exist a function $f_{\xi }\in P{W}_{p,w}^{\Omega }$ such that

$$f_{\xi }\left({\xi }^{\prime }\right)={\delta }_{\xi ,{\xi }^{\prime }},{\xi }^{\prime }\in \Lambda ,$$

with

$$\parallel f_{\xi }{\parallel }_{p,w}^p\le c{w}_1\left(\xi \right)\le c{\int }_{B(\xi ,1)}w\left(x\right)dx.$$

By Lemma 1, we can take entire functions g in one variable of exponential type not exceeding $\epsilon$, such that $g\left(0\right)=1$ and

$$g\left(t\right)\asymp {(1+|t\left|\right)}^{-l},\mathrm{for}\mathrm{all}t\in \mathbb{R},$$

where l is a sufficient large positive integer. Given a sequence ${\left\{c_{\xi }\right\}}_{\xi \in \Lambda }$ satisfying

$$\sum _{\xi \in \Lambda }{\left|c_{\omega }\right|}^p{w}_1\left(\xi \right)<\infty ,$$

we can construct the entire function

$$f\left(z\right)=\sum _{\xi \in \Lambda }{c}_{\xi }{f}_{\xi }\left(z\right)g\left(\right|z-\xi \left|\right),$$

which satisfies $f\left(\xi \right)=c_{\xi },\xi \in \Lambda$ and $\mathrm{supp}\mathcal{F}f\subset {\Omega }_{\epsilon }$. To show that $\Lambda$ is $L_{q,\tilde{w}}$-interpolating for $P{W}_{q,\tilde{w}}^{{\Omega }_{\epsilon }}$, it suffices to prove that for $0<q<\infty ,$

$${\parallel f\parallel }_{q,\tilde{w}}^q\le c\sum _{\xi \in \Lambda }{\left|c_{\xi }\right|}^p{\tilde{w}}_1\left(\xi \right).$$

By Lemma 2, we obtain that $\Lambda$ is uniformly separated. This means that

$$\underset{x\in {\mathbb{R}}^d}{sup}\sum _{\xi \in \Lambda }{\chi }_{B(\xi ,1)}\left(x\right)\le c.$$

It then follows that

$$\begin{array}{cc}\hfill & \underset{x\in {\mathbb{R}}^d}{sup}\sum _{\xi \in \Lambda }g\left(\right|x-\xi \left|\right)\le \underset{x\in {\mathbb{R}}^d}{sup}\sum _{\xi \in \Lambda }{\int }_{B(\xi ,1)}g\left(\right|x-y\left|\right)dy\hfill \\ & \le \underset{x\in {\mathbb{R}}^d}{sup}{\int }_{B(\xi ,1)}g\left(\right|x-y\left|\right)dy\le c.\hfill \end{array}$$

Hence, for $1\le q<\infty$, by Hölder’s inequality and (2), we have

$$\begin{array}{cc}& {\left|f\left(z\right)\right|}^q{\tilde{w}}_1\left(z\right)\le {\left(\sum _{\xi \in \Lambda }|c_{\xi }\left|\right|f_{\xi }\left(z\right)\left|g\right(|z-\xi \left|\right)\right)}^q{\tilde{w}}_1\left(z\right)\hfill \\ \hfill & \le \left(\sum _{\xi \in \Lambda }|c_{\xi }{|}^q|f_{\xi }{\left(z\right)|}^{q}g\left(\right|z-\xi \left|\right)\right){\left(\sum _{\xi \in \Lambda }g\left(\right|z-\xi \left|\right)\right)}^{q/q^{\prime }}{\tilde{w}}_1\left(z\right)\hfill \\ \hfill & \le c\sum _{\xi \in \Lambda }|c_{\xi }{|}^q|f_{\xi }{\left(z\right)|}^q{(1+|z-\xi \left|\right)}^{-l+\tilde{s}}{\tilde{w}}_1\left(\xi \right),\hfill \end{array}$$

where $q^{\prime }$ is the conjugate exponent of q.

Integrating with respect to $z\in {\mathbb{R}}^d$ in both hands and noticing that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^d}|f_{\xi }{\left(z\right)|}^q{(1+|z-\xi \left|\right)}^{-l+\tilde{s}}dz\hfill \\ & \le {\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(\xi \right)\right)}^q{\int }_{{\mathbb{R}}^d}{(1+|z-\xi \left|\right)}^{-l+\tilde{s}+2sq/p}dz\hfill \\ & \le c{\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(\xi \right)\right)}^q,\hfill \end{array}$$

we obtain that

$${\parallel f\parallel }_{q,{\tilde{w}}_1}^q\le c\sum _{\xi \in \Lambda }|c_{\xi }{|}^q{\tilde{w}}_1\left(\xi \right){\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(\xi \right)\right)}^q\le c\sum _{\xi \in \Lambda }{\left|c_{\xi }\right|}^q{\tilde{w}}_1\left(\xi \right)$$

where, for the last inequality, we use the estimate

$$\begin{array}{cc}\hfill {\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(\xi \right)\right)}^q& \le c{\displaystyle \frac{{\int }_{B(\xi ,1)}{\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(x\right)\right)}^{q}w\left(x\right)dx}{{\int }_{B(\xi ,1)}w\left(x\right)dx}}\hfill \\ & \le c{\displaystyle \frac{{\int }_{{\mathbb{R}}^d}{\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(x\right)\right)}^{q}w\left(x\right)dx}{{\int }_{B(\xi ,1)}w\left(x\right)dx}}\hfill \\ & \le c{\displaystyle \frac{\parallel f_{\xi }{\parallel }_{q,w}^q}{w_1\left(\xi \right)}}\le c.\hfill \end{array}$$

By Theorem 1, we obtain

$${\parallel f\parallel }_{q,\tilde{w}}^q\le {\parallel f\parallel }_{q,{\tilde{w}}_1}^q\le c\sum _{\xi \in \Lambda }{\left|c_{\xi }\right|}^q{\tilde{w}}_1\left(\xi \right).$$

Next, we consider the case $0<q<1$. Similarly, we have

$$\begin{array}{cc}& {\left|f\left(z\right)\right|}^q{\tilde{w}}_1\left(z\right)\le {\left(\sum _{\xi \in \Lambda }|c_{\xi }\left|\right|f_{\xi }\left(z\right)\left|g\right(|z-\xi \left|\right)\right)}^q{\tilde{w}}_1\left(z\right)\hfill \\ \hfill & \le c\sum _{\xi \in \Lambda }|c_{\xi }{|}^q|f_{\xi }{\left(z\right)|}^q{(1+|z-\xi \left|\right)}^{-ql+\tilde{s}}{\tilde{w}}_1\left(\xi \right),\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{q,\tilde{w}}^q\le {\parallel f\parallel }_{q,{\tilde{w}}_1}^q& \le c\sum _{\xi \in \Lambda }|c_{\xi }{|}^q{\tilde{w}}_1\left(\xi \right){\left({\left(f_{\xi }\right)}_{2/p,1}^{*}\left(\xi \right)\right)}^q\le c\sum _{\xi \in \Lambda }{\left|c_{\xi }\right|}^q{\tilde{w}}_1\left(\xi \right).\hfill \end{array}$$

The proof of Lemma 3 is complete. □

Lemma 4. 

Let $0<p,q<\infty$, w and $\tilde{w}$ be two doubling weights, and let Λ be an uniformly separated $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$. Then, for $\epsilon >0$, with $|{\Omega }_{-\epsilon }|>0$, Λ is an $L_{q,\tilde{w}}$-sampling set for $P{W}_{q,\tilde{w}}^{{\Omega }_{-\epsilon }}$.

Proof. 

Since $\Lambda$ is uniformly separated, for any $f\in P{W}_{q,w}^{{\Omega }_{-\epsilon }}$, by the definition and property of $f_{\beta ,\sigma }^{*}$, and Theorem 1, we obtain that

$$\begin{array}{cc}\hfill \sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^q{\tilde{w}}_1\left(\xi \right)& \le \sum _{\xi \in \Lambda }|f_{\beta ,\sigma }^{*}{\left(\xi \right)|}^q{\tilde{w}}_1\left(\xi \right)\le C\sum _{\xi \in \Lambda }{\int }_{B(\xi ,1/\sigma )}{\left|f_{\beta ,\sigma }^{*}\left(x\right)\right|}^q\tilde{w}\left(x\right)dx\hfill \\ \hfill & \le C{\int }_{{\mathbb{R}}^d}|f_{\beta ,\sigma }^{*}{\left(x\right)|}^q\tilde{w}\left(x\right)dx\le C{\parallel f\parallel }_{q,\tilde{w}}^q.\hfill \end{array}$$

It suffices to prove that, for any $f\in P{W}_{q,\tilde{w}}^{{\Omega }_{-\epsilon }}$,

$${\parallel f\parallel }_{q,\tilde{w}}^q\le c\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^q{\tilde{w}}_1\left(\xi \right).$$

(16)

First, we consider the case $p\ge 1$, by (8), we have for any $f\in P{W}_{q,\tilde{w}}^{{\Omega }_{-\epsilon }}$,

$$\begin{array}{cc}\hfill {\left|f\left(x\right)\right|}^q& \le {\left({\int }_{{\mathbb{R}}^d}\left|f\left(y\right)\right|K_1(x-y)|dy\right)}^q\hfill \\ & \le {\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|(1+|x-y\left|\right)}^{-l}dy\right)}^q\hfill \\ & \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{(1+|x-y\left|\right)}^{-l}dy\right)}^{q/p}{\left({\int }_{{\mathbb{R}}^d}{(1+|x-y\left|\right)}^{-l}dy\right)}^{q/p^{\prime }}\hfill \\ & \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{(1+|x-y\left|\right)}^{-l}dy\right)}^{q/p},\hfill \end{array}$$

where $p^{\prime }$ is the conjugate exponent of p.

Hence, it follows from (2) that

$$\begin{array}{cc}\hfill {\left|f\left(x\right)\right|}^q{\tilde{w}}_1\left(x\right)& \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{(1+|x-y\left|\right)}^{-l}dy\right)}^{q/p}{\tilde{w}}_1\left(x\right)\hfill \\ & \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{\tilde{w}}_1{\left(y\right)}^{p/q}{w}_1\left(y\right){\left(w_1\left(y\right)\right)}^{-1}{(1+|x-y\left|\right)}^{-l+p\tilde{s}/q}dy\right)}^{q/p}.\hfill \end{array}$$

By Lemma 1, we know there exist the entire functions $g_1\left(y\right)$, $g_2\left(y\right)$, and $g_x\left(y\right)$ of exponential type not exceeding $\epsilon /3$ such that, for any $x,y\in {\mathbb{R}}^d$,

$${\left(w_1\left(y\right)\right)}^{-1}\asymp {\left(g_1\left(y\right)\right)}^p,{\tilde{w}}_1{\left(y\right)}^{1/q}\asymp g_2\left(y\right)$$

(17)

and

$${(1+|x-y\left|\right)}^{-l+p\tilde{s}/q}\asymp {\left(g_x\left(y\right)\right)}^p.$$

(18)

Then, by (17), (18), and the fact that $\Lambda$ is an $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$, we have

$$\begin{array}{cc}\hfill {\left|f\left(x\right)\right|}^q{\tilde{w}}_1\left(x\right)& \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{\left(g_2\left(y\right)\right)}^p{w}_1\left(y\right){\left(g_1\left(y\right)\right)}^p{\left(g_x\left(y\right)\right)}^{p}dy\right)}^{q/p}\hfill \\ & \le c{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{\left(g_2\left(\xi \right)\right)}^p{\left(g_1\left(\xi \right)\right)}^p{\left(g_x\left(\xi \right)\right)}^p\left(w_1\left(\xi \right)\right)\right)}^{q/p},\hfill \end{array}$$

where $supp\mathcal{F}\left(f{g}_1{g}_2{g}_x\right)\subset \Omega .$ Again, by (17) and (18), we obtain that

$${\left|f\left(x\right)\right|}^q{\tilde{w}}_1\left(x\right)\le c{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{\tilde{w}}_1{\left(\xi \right)}^{p/q}{(1+|x-\xi \left|\right)}^{-l+p\tilde{s}/q}\right)}^{q/p}.$$

If $q\ge p$, then, by Hölder inequality, we obtain

$$\begin{array}{cc}\hfill {\left|f\left(x\right)\right|}^q{\tilde{w}}_1\left(x\right)& \le c\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^q{\tilde{w}}_1\left(\xi \right){(1+|x-\xi \left|\right)}^{-l+p\tilde{s}/q}\right){\left(\sum _{\xi \in \Lambda }{(1+|x-\xi \left|\right)}^{-l+p\tilde{s}/q}\right)}^{{\displaystyle \frac{q}{p}}(1-{\displaystyle \frac{p}{q}})}\hfill \\ & \le c\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^q{\tilde{w}}_1\left(\xi \right){(1+|x-\xi \left|\right)}^{-l+p\tilde{s}/q}.\hfill \end{array}$$

Integrating with respect to $x\in {\mathbb{R}}^d$ in both hands, we obtain (16).

Similarly, if $q<p$, then we have

$${\left|f\left(x\right)\right|}^q{\tilde{w}}_1\left(x\right)\le c\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^q{\tilde{w}}_1\left(\xi \right){(1+|x-\xi \left|\right)}^{(-l+{\displaystyle \frac{p\tilde{s}}{q}}){\displaystyle \frac{q}{p}}}.$$

Integrating with respect to $x\in {\mathbb{R}}^d$ in both hands, we obtain (16).

Next, we show that, if $0<p<1$, then $\Lambda$ is an $L_2$-sampling set for $P{W}_2^{{\Omega }_{-\epsilon }}$. In fact, similarly to (17) and (18), we know there exist the entire functions $g_1\left(y\right)$ and $g_x\left(y\right)$ of exponential type not exceeding $\epsilon /2$ such that, for any $x,y\in {\mathbb{R}}^d$,

$${\left(w_1\left(y\right)\right)}^{-1}\asymp {\left(g_1\left(y\right)\right)}^p,\mathrm{and}{(1+|x-y\left|\right)}^{-l}\asymp {\left(g_x\left(y\right)\right)}^p.$$

Then, for any $f\in P{W}_2^{{\Omega }_{-\epsilon }}$, we have $f{g}_1{g}_x\in L_{p,w_1}$ and

$$\begin{array}{cc}\hfill {\left|f\left(x\right)\right|}^2& \le c{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|(1+|x-y\left|\right)}^{-l}dy\right)}^2\hfill \\ & \le c\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{(1+|x-y\left|\right)}^{-l}dy\right)\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^{2-p}{(1+|x-y\left|\right)}^{-l}dy\right)\hfill \\ & \le c\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{w}_1\left(y\right){\left(g_1\left(y\right)\right)}^p{\left(g_x\left(y\right)\right)}^{p}dy\right)\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^{2-p}{(1+|x-y\left|\right)}^{-l}dy\right)\hfill \\ & \le c\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^p{w}_1\left(\xi \right){\left(g_1\left(\xi \right)\right)}^p{\left(g_x\left(\xi \right)\right)}^p\right)\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^{2-p}{(1+|x-y\left|\right)}^{-l}dy\right)\hfill \\ & \le c{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2{(1+|x-\xi \left|\right)}^{-l}\right)}^{p/2}{\left(\sum _{\xi \in \Lambda }{(1+|x-\xi \left|\right)}^{-l}\right)}^{(2-p)/2}\hfill \\ & {\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^2{(1+|x-y\left|\right)}^{-l}dy\right)}^{(2-p)/2}{\left({\int }_{{\mathbb{R}}^d}{(1+|x-y\left|\right)}^{-l}dy\right)}^{p/2}\hfill \\ & \le c{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2{(1+|x-\xi \left|\right)}^{-l}\right)}^{p/2}{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^2{(1+|x-y\left|\right)}^{-l}dy\right)}^{(2-p)/2}.\hfill \end{array}$$

Integrating with respect to $x\in {\mathbb{R}}^d$, we obtain that

$$\begin{array}{cc}\hfill {\parallel f\parallel }_2^2& \le c{\int }_{{\mathbb{R}}^d}{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2{(1+|x-\xi \left|\right)}^{-l}\right)}^{p/2}{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^2{(1+|x-y\left|\right)}^{-l}dy\right)}^{(2-p)/2}dx\hfill \\ & \le c{\left({\int }_{{\mathbb{R}}^d}\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2{(1+|x-\xi \left|\right)}^{-l}dx\right)}^{p/2}{\left({\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^p{(1+|x-y\left|\right)}^{-l}dydx\right)}^{(2-p)/2}\hfill \\ & \le c{\left(\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2\right)}^{p/2}{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(y\right)\right|}^{2}dy\right)}^{(2-p)/2}.\hfill \end{array}$$

Thus, we obtain

$${\parallel f\parallel }_2^2\le c\sum _{\xi \in \Lambda }{\left|f\left(\xi \right)\right|}^2.$$

The proof of Lemma 4 is complete. □

3.2. Proof of Main Result

Proof. of Theorem 3

(i) Since $\Lambda$ is an $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$, it follows from Corollary 1 that there exists a uniformly separated subset ${\Lambda }^{\prime }\subset \Lambda$ which is also an $L_{p,w}$-sampling set for $P{W}_{p,w}^{\Omega }$. By Lemma 4, for arbitrary small $\epsilon >0$, ${\Lambda }^{\prime }$ is an $L_{q,\tilde{w}}$-sampling set for $P{W}_{q,\tilde{w}}^{{\Omega }_{-\epsilon }}$ with $q=2,\tilde{w}=1.$ Then, by Landau’s classical result ([2], Theorem 3)

$$D^{-}\left({\Lambda }^{\prime }\right)\ge {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}}.$$

(ii) Since $\Lambda$ is an $L_{p,w}$-interpolating set for $P{W}_{p,w}^{\Omega }$, by Lemma 2, $\Lambda$ is uniformly separated. It follows from Lemma 3 that for arbitrary small $\epsilon >0$, $\Lambda$ is an $L_{q,\tilde{w}}$-sampling set for $P{W}_{q,\tilde{w}}^{{\Omega }_{\epsilon }}$ with $q=2,\tilde{w}=1.$ Then, by Landau’s classical result ([2], Theorem 4)

$$D^{+}\left({\Lambda }^{\prime }\right)\le {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}}.$$

□

4. Conclusions

In this paper, we established that necessary density conditions for the sampling and interpolation of entire functions in weighted Paley–Wiener spaces with doubling weights. Our main results generalize Landau’s classical necessary density conditions for the unweighted Paley–Wiener spaces to the weighted setting. The key contributions of this work can be summarized as follows:

(i) We proved that, for any $L_{p,w}$-sampling set $\Lambda$ of $P{W}_{p,w}^{\Omega }$, there exists a uniformly separated subsequence ${\Lambda }^{\prime }\subset \Lambda$ satisfying $D^{-}\left({\Lambda }^{\prime }\right)\ge {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}}.$ This extends Landau’s necessary sampling density condition to doubling weights for all $0<p<\infty$.

(ii) For $L_{p,w}$-interpolating sets, we showed that $\Lambda$ must be uniformly separated and satisfy $D^{+}(\Lambda )\le {\displaystyle \frac{|\Omega |}{{\left(2\pi \right)}^d}},$ generalizing Landau’s interpolation density condition to weighted spaces.

(iii) Our results demonstrate a universality property: the necessary density conditions for sampling and interpolation in weighted Paley–Wiener spaces coincide with those in the unweighted case. This indicates that doubling weights preserve the fundamental density characteristics of sampling and interpolation sets.

These findings open several directions for future research:

(i) Extension to more general weight classes, such as non-doubling weights with polynomial growth, where the current techniques may require significant modifications.

(ii) Study of complete interpolating sequences ($L_{p,w}$-sampling and interpolating sets) in multidimensional weighted spaces, which remains largely unexplored beyond the one-dimensional case.