On multivariable Plancherel–Pólya inequality and truncation error upper bounds in irregular sampling

In the note is shown that for the d-dimensional Bernstein functions class Bσ,dp,p>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\varvec{\sigma },d}^p,\, p>0$$\end{document} the Plancherel–Pólya inequality holds with the constant which equals to the product of the constants occuring in the one-dimensional cases. Related truncation error upper bounds are precised in the irregular sampling restoration of functions in several variables.

when f is coming from certain fixed functions space and T := (t n ) n∈Z d ⊂ R d and J is a finite subset of Z d . In the literature is frequent the estimate type like where · p denotes the usual L p -norm, whilst · ∞ ≡ ess sup|·|. Probably the most famous of them is the Plancherel-Pólya inequality. However, in the one-dimensional case when the input function f is of several variables (see for instance [22,23] and the appropriate references given therein), only the existence of such constant C f ,T ( p) is quoted without exact expression depending on the behaviour of f and the point set T.
In the case d = 1, let f ∈ L p (R), p > 0 be entire function of exponential type σ > 0 and let (t n ) n∈Z be uniformly discrete real sequence, i.e. for which inf n =m |t n − t m | ≥ δ > 0. Then [23,Eq. (76)] where The relation (2) presents the Plancherel-Pólya inequality with the bound (3) given in [23,Eq. (76)]. We point out the another fashion estimate for C f ,T ( p) obtained by Boas [3, p. 153, Eq. (2.4)] in one-dimensional case under different assumptions on T. He reported that for f being of type σ in the case when no interval of unit length contains more than λ of the t n nodes holds [3, p. 153 also see [4,Theorems 4 and 7] and [13].
In turn, quite recently Berestova [2] proved that the best constant C f ,Z (2) 1 in B 2 σ is of the following structure [1, p. 45, Theorem 1] where a signifies the ceiling of a, denoting the least integer greater than or equal to a.
In this note we give an insight into the d-dimensional analogue of the Plancherel-Pólya inequality describing the constant B d (σ , δ, p) denoting by B p σ ,d , p > 0 Bernstein function class of d-variable entire functions f of exponential type coordinatewise at most σ j , σ = (σ 1 , . . . , σ d ) and f restricted to R d simultaneously belongs to L p (R d ).

The structure of Plancherel-Pólya constant in Bernstein spaces
Various multidimensional Plancherel-Pólya inequalities can be found in Triebel's book [27]; also, during last years several additional very far going generalizations of the Plancherel-Pólya inequality for functions of several variables were obtained, among others in [7] and [22]. Unfortunately, no explicit estimates of the Plancherel-Pólya constant appeared neither in these articles, nor in articles referenced therein including [23]. So, the following result responds to this question, explaining the structure of the Placherel-Pólya constant.
where B 1 (σ j , δ j , p), j = 1, . . . , d denotes any constant occuring in the onedimensional Plancherel-Pólya inequality, and Proof Take d = 2; in the case d > 2 the proof will be only the iteration of the twodimensional proving procedure. By assumption it is | f (z)| ≤ C exp σ 1 |y 1 | + σ 2 |y 2 | , 1 The assumption t n = n; n ∈ Z means that δ = 1. Obviously, the reverse claim does not hold.
The constant B d (σ , δ, p) can be generated coordinate-wise by any of the constants B 1 (σ, δ, p) which occur in the one-dimensional Plancherel-Pólya inequalities.
Applying the original inequality from [23], the multivariable Plancherel-Pólya constant becomes According to the Boas' bound (4) we readily write that Another, but mainly simpler fashion constant was found by Nikol'skii, consult his monograph [15, p. 122 when σ k ≤ π .
an integer and σ δ < π. Then we have The direct consequence of (6) is the following implication of Theorem 1.

Paley-Wiener input functions
Let us consider the Paley-

Corollary 3 The multidimensional Plancherel-Pólya inequality (7) there holds in
Proof By the Theorem 1 the d-dimensional Plancherel-Pólya inequality (7) holds for Lindner published an evaluation for C f ,T (2) in one-dimensional case [12, p. 186, Lemma 1]. Precisely, for f ∈ PW 2 σ and the uniformly discrete set of uniformly discrete points (z n ) n∈Z separated as inf n =m |z n − z m | ≥ δ > 0 and with bounded imaginary However, since PW 2 σ ≡ B 2 σ (see [8, p. 53]), Lindner's constant is applicable in the previous section as well.
Now, we point out that as (t n ) is uniformly discrete real sequence, τ = 0. Therefore, we have the following result.
to hold the variant of (1) for a function f of several variables of exponential type and p > 0. It is clear that our assumption upon uniformly discrete structure of T is stronger then (13) in which Δ 2 = d k=1 δ 2 k can be taken. However, the authors' main task in [21] was to establish exact truncation error upper bound in multidimensional irregular WKS sampling expansion theorem where the structure and sharpness of C f ,T ( p) is not discussed or precised in detail.

WKS sampling theorem
The classical Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem has been extended to the case of nonuniform/irregular sampling by numerous authors. For detailed information on the theory and its various applications, we refer to [8,14]. Most known irregular sampling results deal with Paley-Wiener functions which have L 2 (R) restrictions on the real line, see e.g. [9,10,28]. In turn, there are only few explicit truncation error upper bounds in multidimensional WKS restoration in open literature, consult [17][18][19][20][21] where the last two papers deal with the multidimensional sampling. Based on the methods developed in [19][20][21] the authors presented new truncation error upper bounds. Having in mind the results and further discussion in the previous sections concerning the Plancherel-Pólya constant, we precise and systematically expose the truncation error upper bound expressions in the multidimensional irregular sampling in L p -spaces.
The research methodology is manifold and includes a mixture of several ideas. We implement in the irregularly sampled signal reconstruction procedure Yen's model I [29, p. 252], in which only finite number measured/discretized signal values migrate from their theoretically assumed positions, combined with the time-shifted interpolation procedure [17,24], when the migrated samples nodes are contained in a finite interval around the reconstruction time x = (x 1 , . . . , x d ). Next, considering in the irregularly spaced sampling reconstruction procedure of initial signals, the approach of window canonical product kernel by Flornes et al. [6] in which the kernel function is built from the migrated nodes around reconstruction time x in a way similar to the Lagrange interpolation. The article [21] synthesized all these ideas in the general irregular sampling reconstruction result, compare also the appropriate publications by the authors [18,19]. Finally, we can call our method time-shifted multidimensional irregular Lagrange-Yen type sampling reconstruction procedure with window canonical product kernel.
Before we expose the result, we recall the following notations and definitions. Let T = t n = n + h n , h := h n , N = (N 1 , . . . , N d , (14) where G N (x, t) denotes a derivative with respect to t, being the Lagrange type window canonical product, and Then the sampling expansion holds uniformly on each bounded x-subset of R d . Moreover, the series in (17) converges absolutely too.
For the general case of multidimensional irregular sampling with window canonical product sampling function S(x, t n ) we have time-jittered nodes also outside of J x . Non-vanishing time-jitter h outside J x leads to functions G N (x, t) given by formulae different from (15), see [9]. In turn, in irregular sampling applications we would like to approximate f (x) using exclusively it's empirically established, non-theoretical values at measuring sample nodes t n indexed by J x . Therefore, we use the truncated to J x sampling approximation sum with G N (x, t) (such that is given by (15)) even for arbitrary sample nodes outside J x . Under such assumptions the truncation error coincides with truncation error for the case given by (14)- (17). One can see that T N,d ( f ; x) depends on non-vanishing h in J x due to multiplicative form of (17) and (18). Therefore, the multidimensional sampling problems are significantly more complex than the one-dimensional ones.

Truncation error upper bounds
In this section we obtain universal truncation bounds for multidimensional irregular sampling reconstruction procedure.
The most frequently appearing estimate of the truncation error is of the form (19) p, q being a conjugated Hölder pair, i.e. 1/ p + 1/q = 1, p ≥ 1.
To obtain a class of truncation error upper bounds when the decay rate of the initial signal function is not known one operates with the estimate A p ≤ C f ,T ( p) f p where C f ,T ( p) is a suitable absolute constant. Thus, (19) becomes We now should evaluate B q above so, that it vanishes with |J x | → ∞. So, the obtained upper bounds will be universal for wide classes of f (x) and T.  (16). Here and

Remark 2
The function Y J x ( f ; x) does not depend on samples in T\{t n : n ∈ J x } and we assume that outside J x it is h ≡ 0. Therefore, the structure of {t n j : n j / ∈ J x j } becomes uniform, that is there t n j ≡ n j . In this case Theorem 2 guarantees that f (x) admits the representation (17).
Theorem 3 precises the results on convergence rates derived in [21]. Namely, having in mind the Plancherel-Pólya constant's values presented in a systematic way in the sections 1-3. and by the structural result of the Theorem 1, which reads we conclude the precise form of the truncation error upper bounds listed for estimates built using the constants derived by Plancherel and Pólya (3), Boas (4), Norvidas (5), Berestova (6), Nikol'skii (11) and finally Lindner's constant (12). It should be also pointed out that any of the mentioned constants needs additional constraints upon the parameters σ, δ and p. So, the corollaries with the upper bounds for the truncation error follow.
To simplify the exposition we introduce here, and in what follows, the shorthand for the constant (20) as The Plancherel-Pólya constant implies the following result. Switching to the Boas result, we conclude Corollary 6 Let f ∈ L p (R d ), p > 0 be entire function of exponential type σ > 0, in the case when no interval of unit length contains more than λ j , j = 1, . . . , d of the t n 's from T. Then we have

The result by Norvidas confirms
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