Abstract and Applied AnalysisVolume 2011 (2011), Article ID 731249, 8 pagesdoi:10.1155/2011/731249

Research Article

On the Localization of the Riesz Means of Multiple Fourier Series of Distributions

A. A. Rakhimov

Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, 43300 Selangor, Malaysia

Received 1 August 2011; Accepted 13 September 2011

Academic Editor: Gerd Teschke

Copyright © 2011 A. A. Rakhimov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study special partial sums of multiple Fourier series of distributions. We obtain sufficient conditions of summation of Riesz means of Fourier expansions of distributions with compact support.

1. Introduction

Reconstruction of a distribution from its Fourier expansions is one of the recently studied problems in harmonic analysis. Well-defined sequence of partial sums always converges in the weak topology [1, 2]. However, one can study these expansions in classical sense in the domains where a distribution coincides with locally integrable function. From the divergence of Fourier series of the Dirac delta function, it follows that a singularity of the distribution makes significant influence for the convergence in a domain where it is very smooth (equal to zero in this case). For eigenfunction expansions of even piecewise smooth functions, its discontinuity has negative effect on convergence at the point far from singularity sets (Pinsky phenomenon) [3–5].

In the present paper, we consider a localization problem of multiple Fourier series of distributions. Unlike one-dimensional case partial sums of multiple Fourier series can be defined in various ways, such as rectangular, square, and spherical partial sums [1, 2]. Fourier expansions of a singular distribution can be also studied in the classical sense in the domains where it coincides with a regular function [1, 2, 6–13]. We prove a localization theorem for nonspherical partial sums, that is, for Fourier series under summation over domains bounded by level surfaces of elliptic polynomials.

2. Preliminaries and Formulation of the Main Result

We denote —the space of -periodic in each variable, infinitely differentiable on functions, where .

Let denote a multiindex, that is, -dimensional vector with integer nonnegative components and let, where .

The system of seminorms produces a locally convex topology in , where runs over all multiindexes. We denote corresponding locally convex topological space. Let be the space of periodic distributions, that is, the space of continuous linear functionals on .

For any distribution from we define its Fourier coefficients as the action of distribution on the test function , where and ,  -dimensional vector with integer coordinates. Then can be represented by Fourier series which always converges in the weak topology (see, e.g., in [1, 2]).

Consider the following polynomial: where is a positive integer number, and .

Polynomial is a homogeneous of degree , that is, and an elliptic, that is,

Thus, a family of bounded sets enjoying the following properties:(a)for any pairs , there is , such that ,(b).

Let . Then -partial sums of series (2.1) define by equality

For any real , we define the Riesz means of (2.6) by At we obtain partial sums (2.6).

Summability of series (2.1), as well as its regularization (2.7), depends on power of singularity of. In order to classify singularities of distributions, we apply periodic Liouville spaces [14].

In this paper we study the convergence of the Riesz means (2.7) in domains where a distribution coincides with the zero (the localization problem). The main result of this work is to prove the following.

Theorem 2.1. Let , and coincide with zero in . If then uniformly on any compact set

3. Auxiliary Lemmas on Estimations of the Dirichlet Kernel

The Riesz means (2.7) can be written as where acts on by and is the Riesz means of -partial sums of multiple Fourier series of the Dirac Delta Function: Note that, if , then is exactly the Riesz means of the Dirichlet Kernel.

First, we estimate (3.2) in the norm of positive Liouville spaces. In this, we use the relation between the kernel (3.2) and the relevant kernel of Fourier integrals. Such a relation is known as the Poisson Summation Formula. The kernel for the corresponding Fourier integrals can be also described by the same polynomial replacing its argument range from to : where in definition of the domain its range must be changed accordantly.

Following asymptotic formula valid for the kernel (3.3), we obtain Lemma 3.1.

Lemma 3.1. Let .  Then, for and

Note, that, integral operators, corresponding to the kernels and , act in different functional spaces. On the assumptions of Lemma 3.1 it follows that, if satisfies condition (2.8) in Theorem 2.1, then .

Suppose that vanishes near the boundary of the cube . Then the function belongs to and preserves all properties of in the interior of . Conversely, if has its support in the interior of , then, if we shift its graph along the coordinate axes with steps which are multiples of , we get a periodic function , which coincides with on , that is,

Fourier coefficient of and the Fourier transformation of are related by the formula . Thus, comparing this with previous formula, we obtain the Poisson summation formula The Poisson summation formula (3.7) holds, for example, if the function satisfies conditions where is any positive number.

Note that from definition of the kernel it follows that

Equality (3.9) establishes relationships between the Fourier coefficients of the kernel and the Fourier transformation of the kernel . Moreover, from Lemma 3.1 and inequality (2.8), it follows that the kernel satisfies conditions (3.8). Thus, from (3.7), we obtain On the other hand, taking into account (3.9), from the definition of the kernel , we obtain Thus, from (3.10) and (3.11), we obtain Then in (3.12) separating the term we obtain where is defined as Then from Lemma 3.1 we immediately obtain the following lemma.

Lemma 3.2. Let   be an arbitrary small number and ,   for any .   If   satisfies (2.8), then

Lemma 3.2 provides an estimation of the second term in (3.13). Moreover, if , then from (3.4) we obtain an estimation for the first term in (3.13). Thus, we proved the following lemma.

Lemma 3.3. Let  be  an arbitrary small number and .   If   satisfies (2.8), then

We will estimate the kernel in the norm of space. Lemma 3.3 provides an estimation at . If , then we have the following estimation see [15].

Theorem 3.4. Let be a compact set, then uniformly by where is an arbitrary domain in such that .

Then using Stein’s interpolation theorem for analytical family of linear operators [16] with and , we obtain the following.

Lemma 3.5. Let   satisfy (2.8), and let be an arbitrary compact set. Then uniformly by where   is an arbitrary domain in such that .

For any number introduce the following functions (kernels): Note that .

Lemma 3.6. Let ,   where  is  an arbitrary small number, and let   satisfy (2.8). Then for any nonnegative number   the following relation is true.

Proof. If is an integer, then (3.20) follows directly from Lemma 3.3 and the relation If is not an integer, then write , where is an integer and . Then there is a positive function , such that and Then statement of the Lemma 3.6 follows from the relation (3.22) and Lemma 3.3. Lemma 3.6 is proved.

From Lemmas 3.5 and 3.6 we obtain the following.

Lemma 3.7. Let   be an arbitrary compact set,   satisfy (2.8), and . Then uniformly by   the following estimation is valid: where   is an arbitrary domain in such that .

4. Proof of the Main Result

Let a distribution have a compact support and belong to the space , where . Let be an arbitrary compact set from and satisfy (2.8). Then from (3.1) it follows that where means a norm in the space and means a norm in the space , and such that .

Let . Then from inequality and Lemma 3.7 it follows that uniformly by from .

Statement of Theorem 2.1 follows from inequality (4.3).

Acknowledgment

This paper has been supported by Universiti Putra Malaysia under Research University Grant (RUGS) no. 05-01-11-1273RU.

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