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∞(TN)—the space of 2π-periodic in each variable, infinitely differentiable on RN functions, where TN={x∈RN:-π<xj≤π}.

Let γ=(γ1,γ2,…,γN) denote a multiindex, that is, N-dimensional vector with integer nonnegative components and letDγ=D1γ1D2γ2⋯DNγN, where Dj=(1/i)(∂/∂xj).

The system of seminorms Supx∈TN|Dγf(x)| produces a locally convex topology in C∞(TN), where γ runs over all multiindexes. We denote ɛ(TN) corresponding locally convex topological space. Let ɛ′(TN) be the space of periodic distributions, that is, the space of continuous linear functionals on ɛ(TN).

For any distribution f from ɛ′(TN) we define its Fourier coefficients fn as the action of distribution f on the test function (2π)-N/2exp(-inx), where x∈TN and n∈ZN, N-dimensional vector with integer coordinates. Then f can be represented by Fourier seriesf=(2π)-N/2∑n∈ZNfnexp(inx),
which always converges in the weak topology (see, e.g., in [1, 2]).

Consider the following polynomial:Pm(n)=(∑j=1r+1nj2)m+1+(∑j=r+2Nnj2)m(∑j=1Nnj2),
where n=(n1,n2,…,nN)∈ZN,m is a positive integer number, and r=0,1,…,N-1.

Polynomial Pm(n) is a homogeneous of degree 2(m+1), that is, Pm(λ⋅n)=λ2(m+1)⋅Pm(n),
and an elliptic, that is, Pm(n)>0,n≠0.

Thus, a family of bounded sets Λ(λ)={n∈ZN:Pm(n)<λ},λ∈R+,
enjoying the following properties:

for any pairs (λ1,λ2)∈R+×R+, there is λ∈R+, such that Λ(λ1)∪Λ(λ2)⊂Λ(λ),

⋃λ∈R+Λ(λ)=ZN.

Let f∈ɛ′(TN). Then Λ-partial sums of series (2.1) define by equalityEλf(x)=(2π)-N/2∑Λ(λ)fnexp(in⋅x).

For any real s,s≥0, we define the Riesz means of (2.6) byEλsf(x)=(2π)-N/2∑Λ(λ)(1-Pm(n)λ)sfnexp(inx).
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 Lpα(TN),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∈Lp-α(TN)⋂ɛ′(TN),1<p≤2,α>0, and coincide with zero in Ω⊂TN. If
s>max{(N-r-1)(1-1/2m)p+r2,N-12}+α,
then uniformly on any compact set K⊂Ωlimλ→∞Eλsf(x)=0.

3. Auxiliary Lemmas on Estimations of the Dirichlet Kernel

The Riesz means (2.7) can be written asEλsf(x)=〈f,Dλs(x-y)〉,
where f acts on Dλs(x-y) by y and Dλs(x) is the Riesz means of Λ-partial sums of multiple Fourier series of the Dirac Delta Function:Dλs(x)=(2π)-N/2∑Λ(λ)(1-Pm(n)λ)sexp(inx).
Note that, if r=N-1, then Dλs(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 Pm replacing its argument range from n∈ZN to ξ∈RN:Θλs(x)=(2π)-N/2∫Λ(λ)(1-Pm(ξ)λ)sexp(iξ⋅x)dξ,
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∈RN,x=(x′,x′′),x′∈Rr+1,x′′∈RN-r-1,0<δ0<|x′|,μ=λ1/2(m+1). Then, for |x′′|<ɛμ-(1-1/2m),0<ɛ<1/2,and μ→∞Θλs(x)=cμNcos(μ|x′|+(r/2-s)(π/2))(μ|x′|)((r+2)/2)+s+((N-1-r)/2m)×(1+O(1μ|x|)+O(|x′′|μ1-1/2m)).

Note, that, integral operators, corresponding to the kernels Dλs and Θλs, 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 Θλs∈L1.

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

Fourier coefficient of f∈TN and the Fourier transformation of g∈L1(RN) are related by the formula fn=(2π)-N/2ĝ(n). Thus, comparing this with previous formula, we obtain the Poisson summation formula∑n∈ZNg(x+2πn)=(2π)-N/2∑n∈ZNĝ(n)expinx.
The Poisson summation formula (3.7) holds, for example, if the function g satisfies conditions|g(x)|≤(1+|x|)-N-ϵ,|ĝ(ξ)|≤(1+|ξ|)-N-ϵ,
where ϵ is any positive number.

Note that from definition of the kernelΘλs it follows thatΘ̂λs(n)={(2π)-N/2(1-Pm(n)λ)s,ifPm(n)≤λ,0,otherwise.

Equality (3.9) establishes relationships between the Fourier coefficients of the kernel Dλs and the Fourier transformation of the kernel Θλs. Moreover, from Lemma 3.1 and inequality (2.8), it follows that the kernel Θλs satisfies conditions (3.8). Thus, from (3.7), we obtain∑n∈ZNΘλs(x+2πn)=(2π)-N/2∑n∈ZNΘ̂λs(n)expinx.
On the other hand, taking into account (3.9), from the definition of the kernel Dλs, we obtainDλs(x)=(2π)-N/2∑n∈ZNΘ̂λs(n)expinx.
Thus, from (3.10) and (3.11), we obtainDλs(x)=∑n∈ZNΘλs(x+2πn).
Then in (3.12) separating the term n=0 we obtainDλs(x)=Θλs(x)+Θ*,λs(x),
where Θ*,λs(x) is defined asΘ*,λs(x)=∑n∈ZN,n≠0Θλs(x+2πn).
Then from Lemma 3.1 we immediately obtain the following lemma.

Lemma 3.2.

Let ɛ>0 be an arbitrary small number and |xi|≤2π-ɛ, for any i=1,2,3,…,N. If s satisfies (2.8), then
Θ*,λs(x)=O(λ1/2(m+1))N-s-1-(r/2)-((N-1-r)/2m).

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 ɛ>0 be an arbitrary small number and |x′|>ɛ. If s satisfies (2.8), then
Dλs(x)=O(λ1/2(m+1))N-s-1-(r/2)-((N-1-r)/2m).

We will estimate the kernel Dλs(x) in the norm of Lq(TN) 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⊂⊂TN be a compact set, then uniformly by x∈K‖Dλs(x-y)‖L2(F)=O(λ(N-1-2s)/4(m+1)),
where F is an arbitrary domain in TN 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⊂⊂TN be an arbitrary compact set. Then uniformly by x∈K‖Dλs(x-y)‖Lq(F)=O(λ1/2(m+1))N-s-1-(r/2)-((N-1-r)/2m),
where F is an arbitrary domain in TNsuch that F¯⋂K=∅,2≤q≤∞.

For any number τ≥0 introduce the following functions (kernels): Dτ,λs(x)=(2π)-N/2∑Λ(λ)Pmτ(n)(1-Pm(n)λ)sexpinx.
Note that D0,λs(x)=Dλs(x).

Lemma 3.6.

Let |x′|>ɛ, where ɛ>0 is an arbitrary small number, and let s satisfy (2.8). Then for any nonnegative number τ the following relation is true.
Dτ,λs(x)-λτDλs(x)=O(1)λ(N-s-2-(r/2)-((N-1-r)/2m)/2(m+1))+τ.

Proof.

If τ is an integer, then (3.20) follows directly from Lemma 3.3 and the relation
λ-1Dk+1,λs(x)=Dk,λs(x)-Dk,λs+1(x).
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≤consttδ-1 and
λ-δDk+1,λs(x)=Dk,λs(x)-δDk,λs+1(x)+∫01Dk,tλs+1(x)ρ(t)dt.

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⊂⊂TN be an arbitrary compact set, s satisfy (2.8), and 2≤q≤∞. Then uniformly by x∈K the following estimation is valid:
‖Dτ,λs(x-y)‖Lq(F)=O(λ(N-s-2-(r/2)-((N-1-r)/2m)/2(m+1))+τ),
where F is an arbitrary domain in TNsuch that F¯⋂K=∅.

4. Proof of the Main Result

Let a distribution f have a compact support and belong to the space Lp-α(TN), where 1<p≤2,α>0. Let K be an arbitrary compact set from TN∖suppf and s satisfy (2.8). Then from (3.1) it follows that|Eλsf(x)|≤‖f‖-α,p‖Dλs(x-y)‖α,q,F,
where ∥·∥-α,p means a norm in the space Lp-α(TN) and ∥·∥α,q,F means a norm in the space Lqα(F),1/q=1-(1/p), and suppf⊂F such that F¯⋂K=∅.

Let τ=α/2(m+1). Then from inequality ‖Dλs‖α,q≤c‖Dτ,λs‖Lq
and Lemma 3.7 it follows thatEλsf(x)=O(1)‖f‖-α,p
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.

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