Abdumalik Rakhimov

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

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.

On the spectral expansions of distributions connected with Schrödinger operators

Abstract In this work the spectral expansions of the distributions connected with Schrodinger’s operator are investigated. The localization principle of Riesz means E λ s f ( x ) of spectral expansions of distributions f from the Sobolev space W 2 − l ( R N ) , l > 0 , is proved for when s ≥ ( N − 1 ) / 2 + l .

Uniform convergence of the eigenfunction expansions of distributions associated with the polyharmonic operator on closed domain

In the solving the boundary value problems of equations of mathematical physics in bounded domains it is important to adjust the solution on the closed domain. This leads to the investigation of convergence and sumability problems near the boundary of the domain. In the present paper we study this problem for the eigenfunction expansions associated with the polyharmonic operator.

2 point block backward difference method for solving Riccati type differential problems

A two point block backward difference method is established to solve Riccati differential equations directly. Based on a predictor-corrector two point block backward difference method (2PBBD), a code is developed using a set of integration coefficients that eliminates the need to be calculated at every step change. The method requires calculating the integration coefficients only once in the beginning. The 2PBBD has an added advantage of a recurrence relationship between coefficients of different orders which provides a more elegant algorithm. The recurrence relationship between coefficients also reduces the computational cost.

On the equiconvergence of the Fourier series and the Fourier integral of distributions

In this paper we prove precise equiconvergence relation between index of the Bochner-Riesz means of the expansions and power of the singularity of the distributions with compact support.

On the uniform convergence of the eigenfunction expansions

We study sufficient conditions for uniform convergence of eigenfunction expansions associated with Schrodingers operator.

On equiconvergence of Fourier Series and Fourier Integral

In this paper we prove a precise equiconvergence relation between index of the Bochner-Riesz means of the expansions and power of the singularity of the distributions with compact support.

On the sufficient conditions of the localization of the Fourier-Laplace series of distributions from liouville classes

In this work we investigate the localization principle of the Fourier-Laplace series of the distribution. Here we prove the sufficient conditions of the localization of the Riesz means of the spectral expansions of the Laplace-Beltrami operator on the unit sphere.

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.

Localization of the Spectral Expansions Associated with the Partial Differential Operators

In this paper we discuss precise conditions of the summability and localization of the spectral expansions associated with various partial differential operators. In this we study the problems in the spaces of both smooth functions and singular distributions. We study spectral expansions of the distributions with the compact support and classify the distributions with the Sobolev spaces. All theorems are formulated in terms of the smoothness and degree of the regularizations.