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

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 𝐶^∞(𝑇^𝑁)—the space of 2𝜋-periodic in each variable, infinitely differentiable on 𝑅^𝑁 functions, where 𝑇^𝑁={𝑥∈𝑅^𝑁∶−𝜋<𝑥_𝑗≤𝜋}.

Let 𝛾=(𝛾₁,𝛾₂,…,𝛾_𝑁) denote a multiindex, that is, 𝑁-dimensional vector with integer nonnegative components and let𝐷^𝛾=𝐷^𝛾1₁𝐷^𝛾2₂⋯𝐷^𝛾𝑁_𝑁, where 𝐷_𝑗=(1/𝑖)(𝜕/𝜕𝑥_𝑗).

The system of seminorms Sup_𝑥∈𝑇^𝑁|𝐷^𝛾𝑓(𝑥)| produces a locally convex topology in 𝐶^∞(𝑇^𝑁), where 𝛾 runs over all multiindexes. We denote 𝜀(𝑇^𝑁) corresponding locally convex topological space. Let 𝜀^′(𝑇^𝑁) be the space of periodic distributions, that is, the space of continuous linear functionals on 𝜀(𝑇^𝑁).

For any distribution 𝑓 from 𝜀^′(𝑇^𝑁) we define its Fourier coefficients 𝑓_𝑛 as the action of distribution 𝑓 on the test function (2𝜋)^−𝑁/2exp(−𝑖𝑛𝑥), where 𝑥∈𝑇^𝑁 and 𝑛∈𝑍^𝑁,  𝑁-dimensional vector with integer coordinates. Then 𝑓 can be represented by Fourier series𝑓=(2𝜋)^−𝑁/2_{𝑛∈𝑍𝑁}𝑓_𝑛exp(𝑖𝑛𝑥), (2.1) which always converges in the weak topology (see, e.g., in [1, 2]).

Consider the following polynomial:𝑃_𝑚(𝑛)=^𝑟+1_𝑗=1𝑛²_𝑗^𝑚+1+^𝑁_𝑗=𝑟+2𝑛²_𝑗^𝑚𝑁_𝑗=1𝑛²_𝑗, (2.2) where 𝑛=(𝑛₁,𝑛₂,…,𝑛_𝑁)∈𝑍^𝑁,𝑚 is a positive integer number, and 𝑟=0,1,…,𝑁−1.

Polynomial 𝑃_𝑚(𝑛) is a homogeneous of degree 2(𝑚+1), that is, 𝑃_𝑚(𝜆⋅𝑛)=𝜆^2(𝑚+1)⋅𝑃_𝑚(𝑛), (2.3) and an elliptic, that is, 𝑃_𝑚(𝑛)>0,𝑛≠0. (2.4)

Thus, a family of bounded sets Λ(𝜆)=𝑛∈𝑍^𝑁∶𝑃_𝑚(𝑛)<𝜆,𝜆∈𝑅⁺, (2.5) enjoying the following properties:(a)for any pairs (𝜆₁,𝜆₂)∈𝑅⁺×𝑅⁺, there is 𝜆∈𝑅⁺, such that Λ(𝜆₁)∪Λ(𝜆₂)⊂Λ(𝜆),(b)⋃_𝜆∈𝑅+Λ(𝜆)=𝑍^𝑁.

Let 𝑓∈𝜀^′(𝑇^𝑁). Then Λ-partial sums of series (2.1) define by equality𝐸_𝜆𝑓(𝑥)=(2𝜋)^−𝑁/2_Λ(𝜆)𝑓_𝑛exp(𝑖𝑛⋅𝑥). (2.6)

For any real 𝑠,𝑠≥0, we define the Riesz means of (2.6) by𝐸^𝑠_𝜆𝑓(𝑥)=(2𝜋)^−𝑁/2_Λ(𝜆)𝑃1−^𝑚(𝑛)𝜆^𝑠𝑓_𝑛exp(𝑖𝑛𝑥). (2.7) At 𝑠=0 we obtain partial sums (2.6).

Summability of series (2.1), as well as its regularization (2.7), depends on power of singularity of𝑓. In order to classify singularities of distributions, we apply periodic Liouville spaces 𝐿^𝛼_𝑝(𝑇^𝑁),1<𝑝≤∞,𝛼∈𝑅 [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 𝑓∈𝐿_𝑝^−𝛼(𝑇^𝑁)⋂𝜀^′(𝑇^𝑁),1<𝑝≤2,𝛼>0, and coincide with zero in Ω⊂𝑇^𝑁. If ()𝑠>max𝑁−𝑟−1)(1−1/2𝑚𝑝+𝑟2,𝑁−12+𝛼, (2.8) then uniformly on any compact set 𝐾⊂Ωlim_𝜆→∞𝐸^𝑠_𝜆𝑓(𝑥)=0. (2.9)

3. Auxiliary Lemmas on Estimations of the Dirichlet Kernel

The Riesz means (2.7) can be written as𝐸^𝑠_𝜆𝑓(𝑥)=𝑓,𝐷^𝑠_𝜆𝑥−𝑦, (3.1) where 𝑓 acts on 𝐷^𝑠_𝜆(𝑥−𝑦) by 𝑦 and 𝐷^𝑠_𝜆(𝑥) is the Riesz means of Λ-partial sums of multiple Fourier series of the Dirac Delta Function:𝐷^𝑠_𝜆(𝑥)=(2𝜋)^−𝑁/2_Λ(𝜆)𝑃1−^𝑚(𝑛)𝜆^𝑠exp(𝑖𝑛𝑥). (3.2) Note that, if 𝑟=𝑁−1, then 𝐷^𝑠_𝜆(𝑥) 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 𝑃_𝑚 replacing its argument range from 𝑛∈𝑍^𝑁 to 𝜉∈𝑅^𝑁:Θ^𝑠_𝜆(𝑥)=(2𝜋)^−𝑁/2_Λ(𝜆)𝑃1−^𝑚(𝜉)𝜆^𝑠exp(𝑖𝜉⋅𝑥)𝑑𝜉, (3.3) where in definition of the domain Λ(𝜆) its range must be changed accordantly.

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

Lemma 3.1. Let 𝑥∈𝑅^𝑁,𝑥=(𝑥^′,𝑥^′′),𝑥^′∈𝑅^𝑟+1,𝑥^′′∈𝑅^{𝑁−𝑟−1},0<𝛿₀<|𝑥^′|,𝜇=𝜆^1/2(𝑚+1).  Then, for |𝑥^′′|<𝜀𝜇^{−(1−1/2𝑚)},0<𝜀<1/2,and 𝜇→∞Θ^𝑠_𝜆(𝑥)=𝑐𝜇^𝑁cos𝜇𝑥^′+(𝑟/2−𝑠)(𝜋/2)𝜇𝑥^{′((𝑟+2)/2)+𝑠+((𝑁−1−𝑟)/2𝑚)}×11+𝑂𝜇|𝑥|+𝑂𝑥^′′𝜇^1−1/2𝑚. (3.4)

Note, that, integral operators, corresponding to the kernels 𝐷^𝑠_𝜆 and Θ^𝑠_𝜆, act in different functional spaces. On the assumptions of Lemma 3.1 it follows that, if 𝑠 satisfies condition (2.8) in Theorem 2.1, then Θ^𝑠_𝜆∈𝐿₁.

Suppose that 𝑓∈𝐿₁(𝑇^𝑁) vanishes near the boundary of the cube 𝑇^𝑁. Then the function𝑔(𝑥)=𝑓(𝑥),if𝑥∈𝑇^𝑁,0,if𝑥∉𝑇^𝑁, (3.5) belongs to 𝐿₁(𝑅^𝑁) and preserves all properties of 𝑓 in the interior of 𝑇^𝑁. Conversely, if 𝑔∈𝐿(𝑅^𝑁) has its support in the interior of 𝑇^𝑁, then, if we shift its graph along the coordinate axes with steps which are multiples of 2𝜋, we get a periodic function 𝑓∈𝐿₁(𝑇^𝑁), which coincides with 𝑔 on 𝑇^𝑁, that is,𝑓(𝑥)=_{𝑛∈𝑍𝑁}𝑔(𝑥+2𝜋𝑛). (3.6)

Fourier coefficient of 𝑓∈𝑇^𝑁 and the Fourier transformation of 𝑔∈𝐿₁(𝑅^𝑁) are related by the formula 𝑓_𝑛=(2𝜋)^−𝑁/2𝑔(𝑛). Thus, comparing this with previous formula, we obtain the Poisson summation formula_{𝑛∈𝑍𝑁}𝑔(𝑥+2𝜋𝑛)=(2𝜋)^−𝑁/2_{𝑛∈𝑍𝑁}𝑔(𝑛)exp𝑖𝑛𝑥. (3.7) The Poisson summation formula (3.7) holds, for example, if the function 𝑔 satisfies conditions𝑔(𝑥)≤(1+|𝑥|)^{−𝑁−𝜖},𝑔(𝜉)≤1+𝜉^{−𝑁−𝜖}, (3.8) where 𝜖 is any positive number.

Note that from definition of the kernelΘ^𝑠_𝜆 it follows thatΘ^𝑠_𝜆(𝑛)=(2𝜋)^−𝑁/2𝑃1−^𝑚(𝑛)𝜆^𝑠,if𝑃_𝑚(𝑛)≤𝜆,0,otherwise. (3.9)

Equality (3.9) establishes relationships between the Fourier coefficients of the kernel 𝐷^𝑠_𝜆 and the Fourier transformation of the kernel Θ^𝑠_𝜆. Moreover, from Lemma 3.1 and inequality (2.8), it follows that the kernel Θ^𝑠_𝜆 satisfies conditions (3.8). Thus, from (3.7), we obtain_{𝑛∈𝑍𝑁}Θ^𝑠_𝜆(𝑥+2𝜋𝑛)=(2𝜋)^−𝑁/2_{𝑛∈𝑍𝑁}Θ^𝑠_𝜆(𝑛)exp𝑖𝑛𝑥. (3.10) On the other hand, taking into account (3.9), from the definition of the kernel 𝐷^𝑠_𝜆, we obtain𝐷^𝑠_𝜆(𝑥)=(2𝜋)^−𝑁/2_{𝑛∈𝑍𝑁}Θ^𝑠_𝜆(𝑛)exp𝑖𝑛𝑥. (3.11) Thus, from (3.10) and (3.11), we obtain𝐷^𝑠_𝜆(𝑥)=_{𝑛∈𝑍𝑁}Θ^𝑠_𝜆(𝑥+2𝜋𝑛). (3.12) Then in (3.12) separating the term 𝑛=0 we obtain𝐷^𝑠_𝜆(𝑥)=Θ^𝑠_𝜆(𝑥)+Θ^𝑠_∗,𝜆(𝑥), (3.13) where Θ^𝑠_∗,𝜆(𝑥) is defined asΘ^𝑠_∗,𝜆(𝑥)=_{𝑛∈𝑍𝑁,𝑛≠0}Θ^𝑠_𝜆(𝑥+2𝜋𝑛). (3.14) Then from Lemma 3.1 we immediately obtain the following lemma.

Lemma 3.2. Let 𝜀>0  be an arbitrary small number and |𝑥_𝑖|≤2𝜋−𝜀,   for any 𝑖=1,2,3,…,𝑁.   If 𝑠  satisfies (2.8), then Θ^𝑠_∗,𝜆(𝑥)=𝑂𝜆^{1/2(𝑚+1)𝑁−𝑠−1−(𝑟/2)−((𝑁−1−𝑟)/2𝑚)}. (3.15)

Lemma 3.2 provides an estimation of the second term in (3.13). Moreover, if 0<𝛿₀<|𝑥^′|, 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 𝜀>0 be  an arbitrary small number and |𝑥^′|>𝜀.   If 𝑠  satisfies (2.8), then 𝐷^𝑠_𝜆(𝑥)=𝑂𝜆^{1/2(𝑚+1)𝑁−𝑠−1−(𝑟/2)−((𝑁−1−𝑟)/2𝑚)}. (3.16)

We will estimate the kernel 𝐷^𝑠_𝜆(𝑥) in the norm of 𝐿_𝑞(𝑇^𝑁) space. Lemma 3.3 provides an estimation at 𝑞=∞. If 𝑞=2, then we have the following estimation see [15].

Theorem 3.4. Let 𝐾⊂⊂𝑇^𝑁 be a compact set, then uniformly by 𝑥∈𝐾𝐷^𝑠_𝜆𝑥−𝑦_𝐿2(𝐹)=𝑂𝜆^{(𝑁−1−2𝑠)/4(𝑚+1)}, (3.17) where 𝐹 is an arbitrary domain in 𝑇^𝑁 such that 𝐹⋂𝐾=∅.

Then using Stein’s interpolation theorem for analytical family of linear operators [16] with 𝑞=∞ and 𝑞=2, we obtain the following.

Lemma 3.5. Let 𝑠  satisfy (2.8), and let 𝐾⊂⊂𝑇^𝑁 be an arbitrary compact set. Then uniformly by 𝑥∈𝐾𝐷^𝑠_𝜆𝑥−𝑦_{𝐿𝑞(𝐹)}=𝑂𝜆^{1/2(𝑚+1)𝑁−𝑠−1−(𝑟/2)−((𝑁−1−𝑟)/2𝑚)}, (3.18) where 𝐹  is an arbitrary domain in 𝑇^𝑁such that 𝐹⋂𝐾=∅,2≤𝑞≤∞.

For any number 𝜏≥0 introduce the following functions (kernels): 𝐷^𝑠_𝜏,𝜆(𝑥)=(2𝜋)^−𝑁/2_Λ(𝜆)𝑃^𝜏_𝑚(𝑛)𝑃1−^𝑚(𝑛)𝜆^𝑠exp𝑖𝑛𝑥. (3.19) Note that 𝐷^𝑠_0,𝜆(𝑥)=𝐷^𝑠_𝜆(𝑥).

Lemma 3.6. Let |𝑥^′|>𝜀,   where 𝜀>0  is  an arbitrary small number, and let 𝑠  satisfy (2.8). Then for any nonnegative number 𝜏  the following relation is true. 𝐷^𝑠_𝜏,𝜆(𝑥)−𝜆^𝜏𝐷^𝑠_𝜆(𝑥)=𝑂(1)𝜆^{(𝑁−𝑠−2−(𝑟/2)−((𝑁−1−𝑟)/2𝑚)/2(𝑚+1))+𝜏}. (3.20)

Proof. If 𝜏 is an integer, then (3.20) follows directly from Lemma 3.3 and the relation 𝜆⁻¹𝐷^𝑠_𝑘+1,𝜆(𝑥)=𝐷^𝑠_𝑘,𝜆(𝑥)−𝐷^𝑠+1_𝑘,𝜆(𝑥). (3.21) If 𝜏 is not an integer, then write 𝜏=𝑘+𝛿, where 𝑘 is an integer and 𝛿∈(0,1). Then there is a positive function 𝜌(𝑡), such that 𝜌≤const𝑡^𝛿−1 and 𝜆^−𝛿𝐷^𝑠_𝑘+1,𝜆(𝑥)=𝐷^𝑠_𝑘,𝜆(𝑥)−𝛿𝐷^𝑠+1_𝑘,𝜆(𝑥)+¹₀𝐷^𝑠+1_{𝑘,𝑡𝜆}(𝑥)𝜌(𝑡)𝑑𝑡. (3.22)
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 𝐾⊂⊂𝑇^𝑁  be an arbitrary compact set, 𝑠  satisfy (2.8), and 2≤𝑞≤∞. Then uniformly by 𝑥∈𝐾  the following estimation is valid: 𝐷^𝑠_𝜏,𝜆𝑥−𝑦_{𝐿𝑞(𝐹)}=𝑂𝜆^{(𝑁−𝑠−2−(𝑟/2)−((𝑁−1−𝑟)/2𝑚)/2(𝑚+1))+𝜏}, (3.23) where 𝐹  is an arbitrary domain in 𝑇^𝑁such that 𝐹⋂𝐾=∅.

4. Proof of the Main Result

Let a distribution 𝑓 have a compact support and belong to the space 𝐿_𝑝^−𝛼(𝑇^𝑁), where 1<𝑝≤2,𝛼>0. Let 𝐾 be an arbitrary compact set from 𝑇^𝑁⧵supp𝑓 and 𝑠 satisfy (2.8). Then from (3.1) it follows that𝐸^𝑠_𝜆𝑓(𝑥)≤𝑓_−𝛼,𝑝𝐷^𝑠_𝜆𝑥−𝑦_{𝛼,𝑞,𝐹}, (4.1) where ‖⋅‖_−𝛼,𝑝 means a norm in the space 𝐿_𝑝^−𝛼(𝑇^𝑁) and ‖⋅‖_{𝛼,𝑞,𝐹} means a norm in the space 𝐿^𝛼_𝑞(𝐹),1/𝑞=1−(1/𝑝), and supp𝑓⊂𝐹 such that 𝐹⋂𝐾=∅.

Let 𝜏=𝛼/2(𝑚+1). Then from inequality 𝐷^𝑠_{𝜆𝛼,𝑞}≤𝑐𝐷^𝑠_{𝜏,𝜆𝐿𝑞} (4.2) and Lemma 3.7 it follows that𝐸^𝑠_𝜆𝑓(𝑥)=𝑂(1)𝑓_−𝛼,𝑝 (4.3) uniformly by 𝑥 from 𝐾.

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.