On the regularizations Fourier series of distributions

Fourier analysis has many applications in various science and technology. In most problem researchers have to analyze functions (data), which has some singularities. This makes some difficulties in Fourier analysis of singular functional. In these, harmonic analysis in the spaces of distributions can be applied. Recently (see for instance [9]-[11]) interest in spectral expansions of distributions increased and number of research papers were published. Present work it devoted to convergence/summation and regularization of Fourier series of distributions in different topologies. In multidimensional case, convergence essentially depends on methods of summation, i.e. on the definition of partial sums. Even “good” defined partial sums may not supply convergence of Fourier series and in this case, some regularization of the partial sums is required

On the uniform convergence of the eigenfunction expansions

We study sufficient conditions for uniform convergence of eigenfunction expansions associated with Schrodinger’s operator

On the summability of the spectral expansions associated with the elliptic differential operators

In this paper we study the summability problems of the spectral expansions associated with the elliptic partial differential operators in the spaces of distributions. In particular, the problems studied for the Schrodinger operator with the singular potential. For this operator theorems on the summability of the Fourier expansions in the generalised Holder spaces and in the Sobolev spaces of the mixed norm proved. The conditions obtained are accurate in the classes of considered distributions

On the approximation of the function on the unite sphere by the spherical harmonics

It is considered the problems of the approximation of the functions defined on the surface of the unite ball in N dimensional spaces by the harmonic polynomials orthogonal on the surface

Convergence of the FourierLaplace series in the spaces with mixed norm

Solution of some boundary value problems and initial problems in unique ball leads to the convergence and summability problems of Fourier series of given function by eigenfunctions of Laplace operator on a sphere - spherical harmonics. Such a series are called as Fourier-Laplace series on sphere. There are a number of works devoted investigation of these expansions in different topologies and for the functions from the various functional spaces. In this paper we study convergence and summability problems of the Fourier Laplace series on the unique sphere in the spaces with the mixed norm

Localization principle of the spectral expansions of distributions connected with Schodinger operator.

In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of distributions are defined. Estimations for Riesz means of the spectral decompositions of the distributions in the norm of the Sobolev classes with negative order are obtained

Localization of fourier-laplace series of distributions

This work was intended as an attempt to extend the results on localization of Fourier-Laplace series to the spectral expansions of distributions on the unit sphere. It is shown that the spectral expansions of the distribution on the unit sphere can be represented in terms of decompostions of Laplace-Beltrami operator. It was of interest to establish sufficient conditions for localization of the spectral expansions of distribution to clarify the latter some relevant counter examples are indicated

Localization principle of the spectral expansions of distributions connected with Schodinger operator.

In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of distributions are defined. Estimations for Riesz means of the spectral decompositions of the distributions in the norm of the Sobolev classes with negative order are obtained