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We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.

Convergence of the Fourier series and integral of integrable functions of one variable at certain point depends only from the values of the function in the small neighbourhood of this point (localizations principles). More- over, the difference of the partial sums of the Fourier series and integral of a function uniformly converge to zero, which means both expansions converge or diverge at the same time (equiconvergence).

In N-dimensional case,

In [

In this paper we study equiconvergence of the Fourier series and integral of the linear continuous functionals (distributions) in the case of spherical summation. Localiation of spectral expansions of distributions for the first time was studied by Sh.A. Alimov [

Let

where K is a compact subset of

Recall

where its Fourier coefficients

The Riesz means of order s,

Now, let us extend f from

where

In this paper we shall be studying a relation between expansions (2) and (3) for some values of the summation index s depending on the power of singularity of f. In fact we will prove uniform equiconvergence of the Riesz means of the Fourier series and the Fourier integral expansion.

However, a behaviour of spherical means for the Fourier series and the Fourier integral expansion can be es

sentially different. The first results on the different behaviour of the Riesz means of critical index

of the Fourier integral and the Fourier series in

lidity of localization principle in

In [

For any real number

Theorem 1 Let

where

Note, if

cides with zero in some neighbourhood of

The illustration of the domains of convergence in the Theorem 1 given in

gence summation domain for the Dirac delta function

Let

Then for any distribution

where f is acting to the test function

Similarly, for the Fourier integral (3) we write

where

Lemma 1 Let

(8). Then

Proof. From the definition of the kernel

Then estimate (9) immediately follows from (11). The estimate (10) follows from (8) and the estimate for the Bessel functions:

Lemma 1 proved.

Note, that if a function

Thus from Lemma 1 applying (12) for the function

Then from (5) and (11) we have

In the sum of right hand side in (13) by separation term

where

Then from Lemma 1 immediately follows:

Lemma 2 Let

From the Formula (15) obtain

Then the statement of the Theorem 1 follows from the lemma below and equality (17):

Lemma 3 Let

Then

uniformly in any compact set

Proof. For any proper domain

where

Note if

where

Then the statement of the Lemma 4 follows from (19) and

Equiconvergence of the Fourier series and integral of distributions depends on singularity of the distribution and power of regularisation as found in the main theorem. Obtained in Theorem 1 a relation for the singularity and summability index is accurate. However, to prove sharp result for the Reisz means below critical index for the smooth functions meets with some difficulties. This circumstance appears due to not applicability of the Poisson formula of summation.

Ongoing research on the topics of the paper supported by IIUM FRGS 14 142 0383.

A. A. Rakhimov, (2015) On the Equiconvergence of the Fourier Series and Integral of Distributions. Journal of Applied Mathematics and Physics,03,1361-1366. doi: 10.4236/jamp.2015.311163