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

Abstract

In the note is shown that for the d-dimensional Bernstein functions class $B_{\varvec{\sigma },d}^p,\, p>0$ 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.

1 Introduction and previous results

In this note we are interested in upper bound estimates for the sum

$$\begin{aligned} A_p = \left( \sum _{{\textbf{n}} \in {\mathbb {Z}}^d{\setminus } {\mathfrak {J}}}|f(t_{{\textbf{n}}})|^p\right) ^{1/p}, \qquad p>0, \end{aligned}$$

when f is coming from certain fixed functions space and $\mathfrak T := (t_{\textbf{n}})_{{\textbf{n}} \in {\mathbb {Z}}^d} \subset {\mathbb {R}}^d$ and ${\mathfrak {J}}$ is a finite subset of ${\mathbb {Z}}^d$. In the literature is frequent the estimate type like

$$\begin{aligned} A_p \le C_{f,{\mathfrak {T}}}(p)\,\Vert f\Vert _p, \qquad 0<p\le \infty \end{aligned}$$

(1)

where $\Vert \cdot \Vert _p$ denotes the usual $L^p$-norm, whilst $\Vert \cdot \Vert _\infty \equiv \mathrm{ess\, \, sup}|\cdot |$. 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 $C_{f,{\mathfrak {T}}}(p)$ is quoted without exact expression depending on the behaviour of f and the point set ${\mathfrak {T}}$.

In the case $d = 1$, let $f \in L^p({\mathbb {R}}),\, p>0$ be entire function of exponential type $\sigma >0$ and let $(t_n)_{n\in {\mathbb {Z}}}$ be uniformly discrete real sequence, i.e. for which $\inf _{n\ne m}|t_n-t_m|\ge \delta >0$. Then [23, Eq. (76)]

$$\begin{aligned} \sum _{n \in {\mathbb {Z}}} |f(t_n)|^p \le {\textsf{B}}_1^{PP}(\sigma , \delta , p)\, \Vert f\Vert _p^p, \end{aligned}$$

(2)

where

$$\begin{aligned} {\textsf{B}}_1^{PP}(\sigma , \delta , p) = \frac{8(e^{p\,\delta \sigma /2} -1)}{ p\, \pi \delta ^2 \sigma } \, . \end{aligned}$$

(3)

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 $C_{f,{\mathfrak {T}}}(p)$ obtained by Boas [3, p. 153, Eq. (2.4)] in one-dimensional case under different assumptions on ${\mathfrak {T}}$. He reported that for f being of type $\sigma$ in the case when no interval of unit length contains more than $\lambda$ of the $t_n$ nodes holds [3, p. 153, Eq. (2.4)]

$$\begin{aligned} {\textsf{B}}_1^{Boas}(\sigma , \delta , p) = 2^{3p+1}\,\frac{\lambda \sigma }{\pi }\, \Big (1+\Big [ \frac{\pi }{2\sigma }\Big ]\Big )\, \Big ( 2+\frac{1}{\pi ^2}\Big )^p\,; \end{aligned}$$

(4)

here, and in what follows [a] denotes for the integral part of some real a.

Let $B_{\sigma }^p,\, 0<p<\infty$ denote the set of entire functions f of exponential type at most $\sigma >0$ for which the restriction $f \big |_{{\mathbb {R}}}$ belongs to $L^p({\mathbb {R}})$. This class of functions we call Bernstein space. The problem of finding the exact form of the Plancherel–Pólya constant $C_{f,{\mathfrak {T}}}(p)$ is well covered for $B_\sigma ^p,\, 1 \le p \le 2$. For instance, Norvidas shows that [16, p. 474, Theorem 1]

$$\begin{aligned} {\textsf{B}}_1^{Nor}(\sigma , \delta , p) = \dfrac{1+[\delta ]}{2 \delta \Vert \cos (\sigma \delta \ x)\Vert ^p_{L^p[0,\frac{1}{2}]}}, \qquad \sigma \,\delta < \pi \,, \end{aligned}$$

(5)

also see [4, Theorems 4 and 7] and [13].

In turn, quite recently Berestova [2] proved that the best constant $C_{f, {\mathbb {Z}}}(2)$¹ in $B_\sigma ^2$ is of the following structure [1, p. 45, Theorem 1]

(6)

where 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 ${\textsf{B}}_d(\varvec{\sigma }, \varvec{\delta }, p)$ denoting by $B_{\varvec{\sigma },d}^p,\, p>0$ Bernstein function class of d-variable entire functions f of exponential type coordinatewise at most $\sigma _j\,, \varvec{\sigma }= (\sigma _1, \ldots , \sigma _d)$ and f restricted to ${\mathbb {R}}^d$ simultaneously belongs to $L^p({\mathbb {R}}^d)$.

2 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.

Theorem 1

Let ${\mathfrak {T}} = \{(t_{{\textbf{n}}}) :{\textbf{n}} \in \mathbb Z^d \}$ be real coordinate-wise increasing, uniformly discrete sequence, i.e. $\inf _{n \ne m}|t_n^{(k)}-t_m^{(k)}|\ge \delta _k>0,\, k = 1,\ldots ,d$, $\varvec{\delta }:= (\delta _1, \ldots , \delta _d)$. Let $f \in B_{\varvec{\sigma }, d}^p,\, p>0$. Then

$$\begin{aligned} \sum _{{\textbf{n}} \in {\mathbb {Z}}^d} |f(t_{{\textbf{n}}})|^p \le \mathsf B_d(\varvec{\sigma }, \varvec{\delta }, p)\,\Vert f\Vert _p^p\,, \end{aligned}$$

(7)

where ${\textsf{B}}_1(\sigma _j, \delta _j, p),\, j = 1,\ldots , d$ denotes any constant occuring in the one-dimensional Plancherel–Pólya inequality, and

$$\begin{aligned} {\textsf{B}}_d(\varvec{\sigma }, \varvec{\delta }, p) = \prod _{j=1}^d {\textsf{B}}_1(\sigma _j, \delta _j, p)\,. \end{aligned}$$

(8)

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

$$\begin{aligned} |f({\textbf{z}})| \le C \exp \big \{ \sigma _1 |y_1|+\sigma _2|y_2| \big \}, \end{aligned}$$

where ${\textbf{z}} = (z_1, z_2);\,z_j = x_j + \textrm{i} y_j,\,j=1,2,$ and because $(t_n^{(j)})_{{\textbf{n}}\in {\mathbb {Z}}^2}$ is separated with $\varvec{\delta }= (\delta _1, \delta _2)$, using (2) we conclude

$$\begin{aligned} \sum _{{\textbf{n}} \in {\mathbb {Z}}^2} |f(t_{{\textbf{n}}})|^p&= \sum _{n_1 \in {\mathbb {Z}}} \Bigg ( \sum _{n_2 \in {\mathbb {Z}}}| f(t_{n_1}^{(1)},t_{n_2}^{(2)})|^p \Bigg ) \nonumber \\&\le {\textsf{B}}_1(\sigma _2, \delta _2, p) \sum _{n_1 \in {\mathbb {Z}}} \int _{{\mathbb {R}}}|f(t_{n_1}^{(1)},x_2)|^p\, \textrm{d}x_2 \nonumber \\&= {\textsf{B}}_1(\sigma _2, \delta _2, p) \int _{{\mathbb {R}}} \Bigg ( \sum _{n_1 \in {\mathbb {Z}}} |f(t_{n_1}^{(1)},x_2)|^p\Bigg ) \textrm{d}x_2 \, . \end{aligned}$$

(9)

Next, the sum $\sum _{n_1 \in {\mathbb {Z}}}|f(t_{n_1}^{(1)},x_2)|^p$ satisfies the conditions needed for the validity of the one-dimensional Plancherel–Pólya inequality (2), hence its second subsequent application to (9) yields

$$\begin{aligned} \sum _{{\textbf{n}} \in {\mathbb {Z}}^2} |f(t_{{\textbf{n}}})|^p&\le {\textsf{B}}_1(\sigma _1, \delta _1, p) {\textsf{B}}_1(\sigma _2, \delta _2, p)\, \int _{{\mathbb {R}}^2} |f(x_1,x_2)|^p\, \textrm{d}x_1 \textrm{d}x_2 \nonumber \\&= {\textsf{B}}_2(\varvec{\sigma }, \varvec{\delta }, p) \Vert f\Vert _p^p\, , \end{aligned}$$

(10)

where the Fubini theorem is applied in (10). The assertion is proved. $\square$

The constant ${\textsf{B}}_d(\varvec{\sigma }, \varvec{\delta }, p)$ can be generated coordinate-wise by any of the constants ${\textsf{B}}_1(\sigma , \delta , p)$ which occur in the one-dimensional Plancherel–Pólya inequalities.

Applying the original inequality from [23], the multivariable Plancherel–Pólya constant becomes

$$\begin{aligned} {\textsf{B}}_d^{PP}(\varvec{\sigma }, \varvec{\delta }, p) = \Big ( \frac{8}{p \pi }\Big )^d \prod _{k=1}^d \frac{e^{p\,\delta _k \sigma _k /2} -1}{ \delta _k^2\sigma _k} \, . \end{aligned}$$

According to the Boas’ bound (4) we readily write that

$$\begin{aligned} {\textsf{B}}_d^{Boas}(\varvec{\sigma }, \varvec{\delta }, p) = 2^d\,\Big (\frac{8^p}{\pi }\Big )^d \Big ( 2+\frac{1}{\pi ^2}\Big )^{p\ d} \prod _{j=1}^d \lambda _j \sigma _j \,\Big (1+\Big [ \frac{\pi }{2\sigma _j}\Big ]\Big )\,. \end{aligned}$$

Another, but mainly simpler fashion constant was found by Nikol’skii, consult his monograph [15, p. 122 et seq., 3.3]

$$\begin{aligned} {\textsf{B}}_d^{Nik}(\varvec{\sigma }, \varvec{\delta }, p) = \prod _{k=1}^d (1+\sigma _k)^p, \qquad 1 \le p <\infty , \end{aligned}$$

(11)

when $\sigma _k \le \pi$.

Corollary 1

Let $f \in B_{\varvec{\sigma }, d}^p;\, 1 \le p \le 2$; $d \ge 2$ an integer and $\sigma \delta < \pi$. Then we have

$$\begin{aligned} {\textsf{B}}_d^{Nor}(\varvec{\sigma }, \varvec{\delta }, p) = \frac{1}{2^d} \prod _{k=1}^d \dfrac{1+[\delta _k]}{\delta _k \Vert \cos (\sigma _k \delta _k\ x)\Vert _{L^p_{[0,1/2]}}}\,. \end{aligned}$$

The direct consequence of (6) is the following implication of Theorem 1.

Corollary 2

Let $f \in B_{\varvec{\sigma }, d}^2$; $d \ge 2$ an integer and $\sigma > \pi$. Then there holds

3 Paley–Wiener input functions

Let us consider the Paley-Wiener space $PW_{\sigma ,d}^p, p>0$ of all complex-valued $L^p({\mathbb {R}}^d)$-functions whose Fourier spectrum is bandlimited to $[-\sigma ,\sigma ]^d$, and $PW_{\sigma ,1}^p \equiv PW_\sigma ^p$.

Corollary 3

The multidimensional Plancherel–Pólya inequality (7) there holds in Paley–Wiener space $PW_{\sigma ,d}^p, p>2$.

Proof

By the Theorem 1 the d-dimensional Plancherel–Pólya inequality (7) holds for functions f from Bernstein space $B_{\sigma ,d}^p,\, p>0$, since $t_{n+1}^{(j)} - t_n^{(j)} \ge \delta _j>0$ for all $j = 1, \ldots , d$. As ${\mathfrak {T}}$ is uniformly discrete exactly in this manner and $B_{\sigma ,d}^p \supseteq PW^q_{\sigma ,d}$, where $p^{-1}+q^{-1} = 1,\,1 < q \le 2$, moreover the spaces $PW_{\sigma ,d}^p$ are decreasingly nested in $p>2$ [8, §6.3–4], it follows that (7) is valid for $PW_{\sigma ,d}^p, p>2$ as well. $\square$

Lindner published an evaluation for $C_{f,{\mathfrak {T}}}(2)$ in one-dimensional case [12, p. 186, Lemma 1]. Precisely, for $f \in PW_\sigma ^2$ and the uniformly discrete set of uniformly discrete points $(z_n)_{n\in {\mathbb {Z}}}$ separated as $\inf _{n \ne m} |z_n-z_m| \ge \delta >0$ and with bounded imaginary parts such that $\sup _n|\Im (z_n)| \le \tau$, there holds

$$\begin{aligned} \sum _{n \in {\mathbb {Z}}} |f(z_n)|^2 \le {\textsf{B}}_1^{Lin}(\sigma , \delta , 2)\, \Vert f\Vert _2^2, \end{aligned}$$

where

$$\begin{aligned} {\textsf{B}}_1^{Lin}(\sigma , \delta , 2) = \dfrac{\textrm{e}^{2\sigma (\tau +1)}-1}{\pi \sigma }\,\Big (1+\dfrac{2}{\delta }\Big )^2\,. \end{aligned}$$

(12)

However, since $PW_\sigma ^2 \equiv B_\sigma ^2$ (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, $\tau = 0$. Therefore, we have the following result.

Corollary 4

Let $f \in PW_{\sigma ,d}^2$ and $d \ge 2$. Then there holds

$$\begin{aligned} {\textsf{B}}_d^{Lin}(\varvec{\sigma }, \varvec{\delta }, 2) = \frac{1}{\pi ^d} \prod _{k=1}^d \dfrac{\textrm{e}^{2\sigma _k}-1}{\sigma _k}\,\Big (1+\dfrac{2}{\delta _k}\Big )^2\,. \end{aligned}$$

Remark 1

In [23, §46, p.148] the authors assume upon ${\mathfrak {T}}$ that

$$\begin{aligned} \sum _{k =1}^d |t_{{\textbf{n}}}^{(k)}-t_{{\textbf{m}}}^{(k)}|^2 \ge \varDelta ^2, \end{aligned}$$

(13)

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 ${\mathfrak {T}}$ is stronger then (13) in which $\varDelta ^2 = \sum _{k=1}^d\delta _k^2$ 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 $C_{f,{\mathfrak {T}}}(p)$ is not discussed or precised in detail. $\square$

4 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 $L^2({\mathbb {R}})$ 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.

5 Multidimensional irregular sampling

The multidimensional irregularly spaced sampling theorem for $B_{\varvec{\sigma },d}^p$ functional class is now presented. The main theorem was proven in [21]. According to Kadec’s 1/4-theorem [11], when the real sequence $(\lambda _n)_{n \in {\mathbb {Z}}}$ for which $|\lambda _n-n| \le L< 1/4, n \in \mathbb Z$, then the system $\big ( \textrm{e}^{\textrm{i} \lambda _n x} \big )_{n \in {\mathbb {Z}}}$ forms a Riesz basis in $L^2[-\pi , \pi ]$. In their publications [25, 26] Sun and Zhu extended and precised Kadec’s result in multidimensional form by Favier and Zalik [5], who gave also certain frame bounds (for $L<1/2$). Sun and Zhu proved that for ${\textbf{n}} = (n_1, \ldots , n_d) \in \mathbb Z^d$ and $\big (\lambda _{{\textbf{n}}}\big )_{{\textbf{n}} \in \mathbb Z^d}$, where

$$\begin{aligned} |\lambda _{n_j} - n_j|<L \le \frac{1}{4}, \qquad j = 1, \ldots , d;\,\, {\textbf{n}} \in {\mathbb {Z}}^d\,, \end{aligned}$$

the system $\big ( \exp \{ \textrm{i} \langle \lambda _j, x_j\rangle \} \big )_{{\textbf{n}} \in {\mathbb {Z}}^d}$ is the Riesz basis in $L^2[-\pi , \pi ]^d$; moreover, the exact frame bound read $(2\pi )^d \big [\cos (\pi L) - \sin (\pi L)\big ]^{2d}$, and $(2\pi )^d \big [2+\cos (\pi L) - \sin (\pi L)\big ]^{2d}$, respectively, consult [25, p. 240, Theorem 2.1].

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 ${\textbf{x}} = (x_1, \ldots , x_d)$. 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 ${\textbf{x}}$ 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 ${\mathfrak {T}} = \big (t_{{\textbf{n}}} = {\textbf{n}} + h_{{\textbf{n}}} \big )$, ${\mathfrak {h}} := \big (h_{{\textbf{n}}}\big )$, ${\textbf{N}}= (N_1, \ldots , N_d)\in {\mathbb {N}}^d$, while

$$\begin{aligned} {\mathfrak {J}}_{{\textbf{x}}} := \Big \{ {\textbf{n}}:\, \bigwedge _{j = 1}^d \big (|x_j - n_j|\le N_j\big ) \Big \}, \end{aligned}$$

and

$$\begin{aligned} S({\textbf{x}}, t_{{\textbf{n}}}) = \prod _{j=1}^d \frac{G_{N_j}(x_j,x_j)}{G_{N_j}'(x_j,t_{n_j})(x_j-t_{n_j})}\,, \end{aligned}$$

(14)

where $G_{N}'(x,t)$ denotes a derivative with respect to t, being

$$\begin{aligned} G_N(x,t) = (t-h_0)\, \textrm{sinc}(t) \mathop {\prod _{|x-k|\le N}}_{k\ne 0} \left( 1- \frac{h_k }{t-k}\right) \frac{k}{t_k}\, , \end{aligned}$$

(15)

the Lagrange type window canonical product, and

$$\begin{aligned} \textrm{sinc}(t) = {\left\{ \begin{array}{ll} \displaystyle \frac{\sin (\pi t)}{\pi t} &{}\quad \hbox {if}\ \ t \ne 0 \\ 1 &{}\quad \hbox {if}\ \ t=0 \end{array}\right. }\,, \end{aligned}$$

stands for the sinc-function. Denote ${\textbf{M}} = (M_1,\ldots ,M_d)$, $\varvec{\delta }=(\delta _1, \ldots ,\delta _d)$, ${\widetilde{M}} = \max \limits _{1 \le j \le d} M_j$, and suppose that $t_{n_j} = n_j + h_{n_j},\, |h_{n_j}|\le M_j$, $j=1,\ldots , d$; for all ${\textbf{n}} \in {\mathfrak {J}}_{\textbf{x}}$.

Theorem 2

[21, p. 599, Theorem 3.1] Let $f\in B^p_{\varvec{\sigma },d},$ $p \ge 1,$ $\sigma _j \le \pi$ for all j,  ${\mathfrak {T}} = \big ( t_{{\textbf{n}}}\big )_{{\textbf{n}} \in {\mathbb {Z}}^d}$ be real separated sequence with

$$\begin{aligned} {{\widetilde{M}}} \le \frac{1}{4}\,\quad \textit{for}\quad p=1 \quad \textit{and} \quad {{\widetilde{M}}}< \frac{1}{4p}\,\quad \textit{for} \quad 1<p<\infty \, . \end{aligned}$$

(16)

Then the sampling expansion

$$\begin{aligned} f({\textbf{x}}) = \sum _{{\textbf{n}} \in {\mathbb {Z}}^d}f(t_{\textbf{n}})\prod _{j=1}^d \frac{G_{N_j}(x_j,x_j)}{G_{N_j}'(x_j,t_{n_j})(x_j-t_{n_j})}, \end{aligned}$$

(17)

holds uniformly on each bounded ${\textbf{x}}$-subset of ${\mathbb {R}}^d$. Moreover, the series in (17) converges absolutely too.

For the general case of multidimensional irregular sampling with window canonical product sampling function $S({\textbf{x}}, t_{\textbf{n}})$ we have time-jittered nodes also outside of $\mathfrak J_{{\textbf{x}}}$. Non-vanishing time-jitter ${\mathfrak {h}}$ outside ${\mathfrak {J}}_{{\textbf{x}}}$ 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 $f({\textbf{x}})$ using exclusively it’s empirically established, non-theoretical values at measuring sample nodes $t_{\textbf{n}}$ indexed by ${\mathfrak {J}}_{{\textbf{x}}}$. Therefore, we use the truncated to ${\mathfrak {J}}_{{\textbf{x}}}$ sampling approximation sum

$$\begin{aligned} Y_{{\mathfrak {J}}_{{\textbf{x}}}}(f;{\textbf{x}})=\sum _{{\textbf{n}} \in {\mathfrak {J}}_{{\textbf{x}}}} f(t_{\textbf{n}}) \prod _{j=1}^d \frac{G_{N_j}(x_j,x_j)}{G_{N_j}'(x_j,t_{n_j})(x_j-t_{n_j})}, \end{aligned}$$

(18)

with $G_{N}(x,t)$ (such that is given by (15)) even for arbitrary sample nodes outside ${\mathfrak {J}}_{{\textbf{x}}}$. Under such assumptions the truncation error

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f;{\textbf{x}})\Vert _\infty =\Vert f({\textbf{x}}) - Y_{{\mathfrak {J}}_{{\textbf{x}}}}(f;{\textbf{x}})\Vert _\infty \end{aligned}$$

coincides with truncation error for the case given by (14)–(17). One can see that $T_{{\textbf{N}},d}(f;\textbf{x})$ depends on non-vanishing ${\mathfrak {h}}$ in ${\mathfrak {J}}_{\textbf{x}}$ due to multiplicative form of (17) and (18). Therefore, the multidimensional sampling problems are significantly more complex than the one-dimensional ones.

6 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

$$\begin{aligned} \Vert T_{{\mathfrak {J}}}(f;{\textbf{x}})\Vert _\infty \le \left( \sum _{{\textbf{n}} \in {\mathbb {Z}}^d{\setminus } {\mathfrak {J}}}| f(t_{{\textbf{n}}})|^p \right) ^{\frac{1}{p}}\, \left( \sum _{{\textbf{n}} \in {\mathbb {Z}}^d{\setminus } {\mathfrak {J}}}| S({\textbf{x}},t_{{\textbf{n}}})|^q\right) ^{\frac{1}{q}} =: A_p\,B_q\, , \end{aligned}$$

(19)

p, q being a conjugated Hölder pair, i.e. $1/p+1/q=1,\, p\ge 1$.

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 $A_p \le C_{f,{\mathfrak {T}}}(p)\,\Vert f\Vert _p$ where $C_{f, {\mathfrak {T}}}(p)$ is a suitable absolute constant. Thus, (19) becomes

$$\begin{aligned} \Vert T_{{\mathfrak {J}}}(f;{\textbf{x}})\Vert _\infty \le B_q\, C_{f,\mathfrak T}(p)\, \Vert f\Vert _p. \end{aligned}$$

We now should evaluate $B_q$ above so, that it vanishes with $|{\mathfrak {J}}_{{\textbf{x}}}| \rightarrow \infty$. So, the obtained upper bounds will be universal for wide classes of $f({\textbf{x}})$ and ${\mathfrak {T}}$.

Theorem 3

[21, p. 600, Theorem 4.1] Let $f\in B^p_{\varvec{\sigma },d},\, p\ge 1$. Then we have

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty \le K_{\varvec{\delta }}({\textbf{N}},{\textbf{M}})\cdot \Vert f\Vert _p\, \end{aligned}$$

where ${\textbf{N}} = (N_1, \ldots , N_d) \in {\mathbb {N}}^d$, whilst ${\widetilde{M}}$ and ${\textbf{M}} = (M_1, \ldots , M_d)$ satisfies (16). Here

$$\begin{aligned} K_{\varvec{\delta }}({\textbf{N}},{\textbf{M}})&= \mathsf B_d^{\frac{1}{p}}(\varvec{\sigma }, \varvec{\delta }, p) \Bigg (\sum _{k=1}^d C_1(N_k,M_k) \nonumber \\&\qquad \cdot \mathop {\prod _{j=1}^d}_{j\ne k} \Big ( C_1(N_j,M_j) + C_2(M_j,\delta )C_3(N_j,M_j)\Big )\Bigg )^{\frac{1}{q}} \,, \end{aligned}$$

(20)

and

$$\begin{aligned} C_1(N,M)&= \frac{2 \pi ^q (2M+1)^q(2N+3)^{2Mq}}{2^{Mq} 3^q\,\varGamma ^{2q}(M+\frac{1}{2})} \frac{q-1+N}{q-1} \left( \frac{(1+N)^{M}(N-\frac{1}{2})}{N(N-M-\frac{1}{2})}\right) ^q,\\ C_2(M,\delta )&= \frac{3^q\, \pi ^{(2M-1)q}\, \textrm{e}^{4Mq}}{2^{(2M+1)q}\, M^{(4M+1)q}\, \delta ^{2Mq}},\\ C_3(N,M)&= \frac{(2M+3)^{(4M-1)q}}{2^{(4M-1)q-1}} \Big (1-\frac{2M+3}{(4M-1)q+1}\Big ) \\&\qquad + \frac{2\left( N+M+\frac{3}{2}\right) ^{(4M-1)q+1}}{(4M-1)q+1} . \end{aligned}$$

Remark 2

The function $Y_{{\mathfrak {J}}_{{\textbf{x}}}}(f;{\textbf{x}})$ does not depend on samples in ${\mathfrak {T}}{\setminus } \{t_{\textbf{n}}:\,{\textbf{n}}\in {\mathfrak {J}}_{{\textbf{x}}}\}$ and we assume that outside ${\mathfrak {J}}_{{\textbf{x}}}$ it is ${\mathfrak {h}} \equiv 0.$ Therefore, the structure of $\{t_{n_j}:\,n_j\not \in {\mathfrak {J}}_{x_j}\}$ becomes uniform, that is there $t_{n_j} \equiv n_j$. In this case Theorem 2 guarantees that $f(\textbf{x})$ admits the representation (17). $\square$

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

$$\begin{aligned} {\textsf{B}}_d(\varvec{\sigma }, \varvec{\delta }, p) = \prod _{k=1}^d {\textsf{B}}_1(\sigma _k, \delta _k, p)\,, \end{aligned}$$

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 $\sigma , \delta$ 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

$$\begin{aligned} {\mathfrak {K}}_{\varvec{\delta }, q}^q({\textbf{N}},{\textbf{M}}) := \sum _{k=1}^d C_1(N_k,M_k) \mathop {\prod _{j=1}^d}_{j\ne k} \Big ( C_1(N_j,M_j) + C_2(M_j,\delta )C_3(N_j,M_j)\Big )\,. \end{aligned}$$

The Plancherel–Pólya constant implies the following result.

Corollary 5

Let $f \in L^p({\mathbb {R}}^d),\, p>0$ be entire function of exponential type $\sigma >0$. Then we have

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty \le \left( \frac{8}{p\,\pi }\right) ^{\frac{d}{p}} \prod _{j=1}^d \left( \frac{e^{p\,\delta _j \sigma _j/2} -1}{\delta _j^2 \sigma _j}\right) ^{\frac{1}{p}} \cdot {\mathfrak {K}}_{\varvec{\delta }, q}({\textbf{N}},{\textbf{M}}) \cdot \Vert f\Vert _p\,. \end{aligned}$$

Switching to the Boas result, we conclude

Corollary 6

Let $f \in L^p({\mathbb {R}}^d),\, p>0$ be entire function of exponential type $\sigma >0$, in the case when no interval of unit length contains more than $\lambda _j, j=1,\ldots , d$ of the $t_{{\textbf{n}}}$’s from ${\mathfrak {T}}$. Then we have

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty&\le 2^{(3+\frac{1}{p})d}\,\frac{(2\pi ^2+1)^d}{\pi ^{(2+\frac{1}{p})d}}\, \left\{ \prod _{j=1}^d \lambda _j\sigma _j\, \left( 1+\left[ \frac{\pi }{2\sigma _j}\right] \right) \right\} ^{\frac{1}{p}} \\&\qquad \cdot {\mathfrak {K}}_{\varvec{\delta }, q}({\textbf{N}},\textbf{M}) \cdot \Vert f\Vert _p\,. \end{aligned}$$

The result by Norvidas confirms

Corollary 7

Let $f \in B_{\varvec{\sigma }, d}^p,\,1 \le p \le 2$ and . Then for $\sigma _j\,\delta _j < \pi ,\, j=1, \ldots , d$

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty \le 2^{-\frac{d}{p}}\,\left\{ \prod _{j=1}^d \dfrac{1+[\delta _j]}{\delta _j \Vert \cos (\sigma _j \delta _j\ x)\Vert ^p_{L^p[0,\frac{1}{2}]}}\right\} ^{\frac{1}{p}} \cdot {\mathfrak {K}}_{\varvec{\delta }, q}({\textbf{N}},{\textbf{M}}) \cdot \Vert f\Vert _p\,. \end{aligned}$$

According to Nikol’skii we have

Corollary 8

Let $f \in B_{{\varvec{\sigma }}, d}^p,\,1 \le p<\infty ;\, \sigma _j < \pi ,\, j=1, \ldots , d$. Then there holds

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty \le 2^{-\frac{d}{p}}\, \prod _{j=1}^d (1+\sigma _j) \cdot {\mathfrak {K}}_{\varvec{\delta }, q}({\textbf{N}},{\textbf{M}}) \cdot \Vert f\Vert _p\,. \end{aligned}$$

On the other hand Berestova’s exact value for the Plancherel–Pólya constant for regularly spaced sampling nodes implies

Corollary 9

Let $f \in B_{\varvec{\sigma }, d}^2;\, \sigma _j >0,\, j=1, \ldots , d$. Then there holds

Finally, we give the result by Lindner, which is related to the Paley–Wiener input functions from $PW_{\sigma ,d}^p, p>2$, compare Corollary 3.

Corollary 10

Let $f \in PW_{{\varvec{\sigma }},d}^2$ and $d \ge 2$. Then

$$\begin{aligned} \Vert T_{{\textbf{N}},d}(f,{\textbf{x}})\Vert _\infty \le \frac{1}{\pi ^{\frac{d}{p}}} \prod _{j=1}^d \dfrac{\left( \textrm{e}^{2\sigma _j}-1\right) ^{\frac{1}{p}}}{\sigma _j^{\frac{1}{p}}}\, \left( 1+\dfrac{2}{\delta _j}\right) ^{\frac{2}{p}} \cdot \mathfrak K_{\varvec{\delta }, q}({\textbf{N}},{\textbf{M}}) \cdot \Vert f\Vert _p\,. \end{aligned}$$

Remark 3

The magnitude evaluations of the upper bounds in corollaries 5–10 still remain when the $\min _{1\le j\le d}(N_j)$ is growing (see [19, p. 570, Theorem 3]), where $p \ge 1;\,\sigma _j \le \pi , j=1, \ldots , d$ was assumed.

Moreover, the results in [21, p. 600, Corollaries 4.2 and 4.3] are modestly more developed under the same assumptions on the initial function f coming from the Paley–Wiener space $PW_{\varvec{\pi }, d}^p;\,p\ge 1$. $\square$