Approximation by multivariate quasi-projection operators and Fourier multipliers

doi:10.1016/j.amc.2021.125955

Applied Mathematics and Computation

Volume 400, 1 July 2021, 125955

https://doi.org/10.1016/j.amc.2021.125955 Get rights and content

Highlights

•
The study of wide classes of multivariate quasi-projection operators.
•
New upper and lower estimates of the approximation error.
•
The results are given in terms of Fourier multipliers.
•
Estimates in terms of anisotropic moduli of smoothness and best approximations.
•
New general Whittaker–Nyquist–Kotelnikov–Shannon-type theorems.

Abstract

Multivariate quasi-projection operators $Q_{j} (f, φ, \tilde{φ}),$ associated with a function $φ$ and a distribution/function $\tilde{φ},$ are considered. The function $φ$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\tilde{φ}$ . Using technique based on the Fourier multipliers, we study approximation properties of such operators for functions $f$ from anisotropic Besov spaces and $L_{p}$ spaces with $1 \leq p \leq \infty$ . In particular, upper and lower estimates of the $L_{p}$ -error of approximation in terms of anisotropic moduli of smoothness and anisotropic best approximations are obtained.

Introduction

The multivariate quasi-projection operator with a matrix dilation $M$ is defined as: $Q_{j} (f, φ, \tilde{φ}) = | \det M |^{j} \sum_{n \in Z^{d}} 〈 f, \tilde{φ} (M^{j} \cdot + n) 〉 φ (M^{j} \cdot + n),$ where $φ$ is a function, $\tilde{φ}$ is a tempered distribution, and $〈 f, \tilde{φ} (M^{j} \cdot + n) 〉$ is an appropriate functional.

The class of operators $Q_{j} (f, φ, \tilde{φ})$ is quite large. It includes the operators associated with a regular function $\tilde{φ},$ in particular, the so-called scaling expansions appearing in wavelet constructions (see, e.g., [3], [11], [12], [20], [21], [28]) as well as the Kantorovich-Kotelnikov operators and their generalizations (see, e.g., [8], [16], [18], [25], [33]). An essentially different class consists of the operators $Q_{j} (f, φ, \tilde{φ})$ associated with a tempered distribution $\tilde{φ}$ related to the Dirac delta-function (the so-called sampling-type operators). The model example of such operators is the following classical sampling expansion, appeared originally in the Kotelnikov formula, $\sum_{n \in Z} f (- 2^{- j} n) \frac{\sin π (2^{j} x + n)}{π (2^{j} x + k)} = 2^{j} \sum_{n \in Z} 〈 f, δ (2^{j} \cdot + n) 〉 sinc (2^{j} x + n),$ where $δ$ is the Dirac delta-function and $sinc x : = \frac{\sin π x}{π x}$ . In recent years, many authors have studied approximation properties of the sampling-type operators for various functions $φ$ (see, e.g., [4], [5], [13], [15], [18], [21], [27], [32]). Consideration of functions $φ$ with a good decay is very useful for different engineering applications. In particular, the operators associated with a linear combination of $B$ -splines as $φ,$ and the Dirac delta-function as $\tilde{φ},$ was studied, e.g., in [2], [6], [27]. For a class of fast decaying functions $φ,$ the sampling-type quasi-projection operators were considered in [21], where the error estimates in the $L_{p}$ -norm, $p \geq 2,$ were obtained in terms of the Fourier transform of $f,$ and the approximation order of the operators was found in the case of an isotropic matrix $M$ . These results were extended to an essentially wider class of functions $φ$ in [7] (see Theorem A below). Next, in the paper [17], the results of [21] were improved in several directions. Namely, the error estimates were obtained also for the case $1 \leq p < 2,$ the requirements on the approximated function $f$ were weakened, and the estimates were given in terms of anisotropic moduli of smoothness and best approximations.

The main goal of the present paper is to extend the results of [17] to band-limited functions $φ$ and to the case $p = \infty$ . The scheme of the proofs of our results is similar to the one given in [17], but the technique is essentially refined by means of using Fourier multipliers. This development allows also to improve the results for the class of fast decaying functions $φ$ and to obtain lower estimates for the $L_{p}$ -error of approximation by quasi-projection operators in some special cases. Similarly, the main result of [16] (see Theorem B below) is essentially extended in several directions (lower estimates, fractional smoothness, approximation in the uniform metric).

The paper is organized as follows. Notation and preliminary information are given in Sections 2 and 3, respectively. Section 4 contains auxiliary results. The main results are presented in Section 5. In particular, the $L_{p}$ -error estimates for quasi-projection operators $Q_{j} (f, φ, \tilde{φ})$ in the case of weak compatibility of $φ$ and $\tilde{φ}$ are obtained in Section 5.2. In this subsection, we also consider lower estimates for the $L_{p}$ -error and a generalization of compatibility conditions to the case of fractional smoothness. Section 5.3 is devoted to approximation by operators $Q_{j} (f, φ, \tilde{φ})$ in the case of strict compatibility of $φ$ and $\tilde{φ}$ . Two generalizations of the Whittaker–Nyquist–Kotelnikov–Shannon-type theorem are also proved in this subsection.

Section snippets

Notation

As usual, we denote by $N$ the set of positive integers, $R^{d}$ is the $d$ -dimensional Euclidean space, $Z^{d}$ is the integer lattice in $R^{d},$ $Z_{+}^{d} : = {x \in Z^{d} : x_{k} \geq 0, k = 1, \dots, d},$ and $T^{d} = R^{d} / Z^{d}$ is the $d$ -dimensional torus. Let $x = (x_{1}, \dots, x_{d})^{T}$ and $ξ = (ξ_{1}, \dots, ξ_{d})^{T}$ be column vectors in $R^{d},$ then $(x, ξ) : = x_{1} ξ_{1} + \dots + x_{d} ξ_{d},$ $| x | : = \sqrt{(x, x)},$ $0 = (0, \dots, 0)^{T} \in R^{d},$ and $B_{δ} = {x \in R^{d} : | x | < δ}$ .

Given $a, b \in R^{d}$ and $α \in Z_{+}^{d},$ we set $[α] = \sum_{k = 1}^{d} α_{k}, D^{α} f = \frac{\partial^{[α]} f}{\partial x^{α}} = \frac{\partial^{[α]} f}{\partial^{α_{1}} x_{1} \dots \partial^{α_{d}} x_{d}}, a^{b} = \prod_{j = 1}^{d} a_{j}^{b_{j}}, α! = \prod_{j = 1}^{d} α_{j}! .$

If $M$ is a $d \times d$ matrix, then $∥ M ∥$ denotes its operator norm in $R^{d}$ ; $M^{*}$ denotes

Preliminary information and main definitions

In what follows, we discuss the operator $Q_{j} (f, φ, \tilde{φ}) : = \sum_{k \in Z^{d}} 〈 f, {\tilde{φ}}_{j k} 〉 φ_{j k},$ where the “inner product” $〈 f, {\tilde{φ}}_{j k} 〉$ has meaning in some sense. This operator is associated with a matrix $M,$ which is a matrix dilation by default.

The expansion $\sum_{k \in Z^{d}} 〈 f, {\tilde{φ}}_{j k} 〉 φ_{j k}$ is an element of the shift-invariant space generated by the function $φ$ . It is known that a function $f$ may be approximated by the elements of such shift-invariant space only if $φ$ satisfies the so-called Strang-Fix conditions.

Definition 1

We say that a function $φ$

Auxiliary results

Lemma 7

[31, Theorem 4.3.1]

Let $g \in L_{p},$ $1 \leq p < \infty,$ and $supp \hat{g} \subset [- σ_{1}, σ_{1}] \times \dots \times [- σ_{d}, σ_{d}],$ $σ_{j} > 0,$ $j = 1, \dots, d$ . Then $\frac{1}{σ_{1} \dots σ_{d}} \sum_{k \in Z^{d}} \max_{x \in Q_{k, σ}} | g (x) |^{p} \leq c {∥ g ∥}_{p}^{p},$ where $Q_{k, σ} = [\frac{2 k_{1} - 1}{2 σ_{1}}, \frac{2 k_{1} + 1}{2 σ_{1}}] \times \dots \times [\frac{2 k_{d} - 1}{2 σ_{d}}, \frac{2 k_{d} + 1}{2 σ_{d}}]$ and $c$ depends only on $p$ and $d$ .

Lemma 8

Let $1 \leq p \leq \infty,$ $g \in L_{p},$ $h \in L_{p^{'}},$ and $\hat{h} \in M_{p}$ . Then the operator $T (g) : = h * g$ is bounded in $L_{p}$ and ${∥ h * g ∥}_{p} \leq ∥ \hat{h} ∥_{M_{p}} {∥ g ∥}_{p}$ .

Proof. In the case $p = \infty,$ the statement is trivial even without the assumption $\hat{h} \in M_{p}$ . Consider the case $p < \infty$ . Choose a sequence ${g_{n}}_{n} \subset S$ converging to $g$ in $L_{p}$ -norm. Since $\hat{h} \in M_{p},$ the functions $Λ_{\hat{h}} (g_{n})$ form a Cauchy

Main lemma

Let $M \in M,$ $α \in A_{M},$ and let $\tilde{φ}$ belong to $S_{α, p; M}^{'}$ . In what follows, we understand $〈 f, {\tilde{φ}}_{j k} 〉$ in the sense of Definition 4. Thus, the quasi-projection operators $Q_{j} (f, φ, \tilde{φ}) = \sum_{k \in Z^{d}} 〈 f, {\tilde{φ}}_{j k} 〉 φ_{j k}$ are defined for all $f \in B_{p; M}^{α (\cdot)}$ . By Lemmas 9 and 13, we have that ${〈 f, {\tilde{φ}}_{j k} 〉}_{k} \in ℓ_{p}$ and ${〈 f, {\tilde{φ}}_{j k} 〉}_{k} \in c_{0}$ if $p = \infty$ . This, together with Lemmas 10 and 11, implies that the series $\sum_{k \in Z^{d}} 〈 f, {\tilde{φ}}_{j k} 〉 φ_{j k}$ converges unconditionally in $L_{p}$ . Therefore, the operator $Q_{j} (f, φ, \tilde{φ})$ is well defined.

An analogue of the following lemma for the case $φ \in L_{p}$

Acknowledgments

The first author was supported by the DFG project KO 5804/1-1. The second author was supported by the RSF project 18-11-00055 (Section 5.3, Lemmas 13, 15 and a half of the proof of Theorem 20 belong to this author).

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Approximation by multivariate quasi-projection operators and Fourier multipliers

Highlights

Abstract

Introduction

Section snippets

Notation

Preliminary information and main definitions

Auxiliary results

[31, Theorem 4.3.1]

Main lemma

Acknowledgments

J. Approx. Theory

J. Approx. Theory

Izv. Math.

Appl. Comput. Harmon. Anal.

Anal. Appl.

J. Approx. Theory

Interpolation Spaces. An Introduction

Prediction by samples from the past with error estimates covering discontinuous signals

IEEE Trans. Inform. Theory

Approximation from shift-invariant subspaces of L2(Rd)

Trans. Am. Math. Soc.

Reconstruction of signals in Lp(R)-space by generalized sampling series based on linear combinations of B-splines

Integral Transforms Spec. Funct.

Quasi-projection operators with applications to differential-difference expansions

Appl. Math. Comput.

Rate of approximation for multivariate sampling kantorovich operators on some functions spaces

J. Int. Eq. Appl.

Strong converse inequalities

J. Anal. Math.

Classical Fourier Analysis

Approximation from shift-invariant subspaces of $L_{2} (R^{d})$

Reconstruction of signals in $L_{p} (R)$ -space by generalized sampling series based on linear combinations of B-splines