Anvarjon Ahmedov

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 .

The estimations for maximal operators

In this paper we study the almost everywhere convergence of the spectral expansions related to the Laplace–Beltrami operator on the unit sphere. Using the spectral properties of the functions with logarithmic singularities, the estimate for maximal operator of the Riesz means of the partial sums of the Fourier–Laplace series is established. We have constructed a different method for investigating the summability problems of Fourier–Laplace series, which based on the theory of spectral decompositions of the self-adjoint Laplace–Beltrami operator.

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

We develop the Newton-Kantorovich method to solve the system of nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

Numerical Integration Of The Integrals Based On Haar Wavelets

In this work, we present a computational method for solving double and triple integrals with variable limits of integrations which is based on Haar wavelets. This approach is the generalization and improvement of the methods [3]. The advantage of this new methods is its more efficient and simple applicability than the previous methods. Error analysis for the case of two dimension are considered. Finally, we also give some numerical examples to compared with existing methods.

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 estimation of the eigenfunctions of biharmonic operator in closed domain

In the current research, the sufficient conditions for uniform convergence of the eigenfunction expansions of the biharmonic operator in closed domain are investigated. The problems which appear in the study of various vibrating systems are the reasons to develop the theory of eigenfunction expansions of the biharmonic operators. Because of the mathematical description of the physical processes taking place in real space are based on the spectral theory of differential operators, particularly biharmonic operator. The biharmonic equation is encountered in plane problems of elasticity. It is also used to describe slow flows of viscous incompressible fluids. In this paper the uniform estimations for eigenfunctions of biharmonic operator is obtained using the resolvent operator of the biharmonic equations. In the future works the obtained estimation will be used to prove uniform convergence of eigenfunction expansions of biharmonic operator in closed domain.

The summability of eigenfunction expansions connected with Schrödinger operator for the functions from Nikolskii classes, H2α(Ω¯) on closed domain

The eigenfunction expansions of the Schrodinger operator on closed domain are considered. The necessary estimations for uniform convergent of the eigenfunction expansions of Schrodinger operator on closed domain are obtained. The sufficient conditions for summability of spectral expansions of continuous functions from Nikolskii classes, H2α(Ω¯) are formulated.

The almost everywhere convergence of the eigenfunction expansions from Liouville classes corresponding to the elliptic operators

Many problems of mathematical physics can be solved by separation methods of partial differential equations. By application of separation method, a solution of the partial differential equation can be represented in terms of eigenfunction expansions of elliptic differential operator. To obtain the solution from its eigenfunction expansion one has to investigate the conditions for convergence of such expansions. In this paper, the problems of almost everywhere convergence of the eigenfunction expansions of the functions from Liouville classes are investigated. The Lebesgue constant corresponding to the elliptic polynomials are estimated. The Jackson Theorem is applied to prove the convergence of multiple Fourier series in the classes of Liouville.

Uniform convergence of the Fourier-Laplace series

This study examines the problem of uniform convergence for the functions from the Nikolskii class. The uniform convergences for the Riesz means of the Fourier-Laplace series in the Nikolskii class Hpa(SN) proved under certain conditions.

Absence of localization of Fourier-Laplace series

This article investigates a function f(x), constructed from the Nikol’skii class in S2. The estimation obtained will show that the Riesz mean of the spectral expansions is unable to be strengthened due to absence of localization caused by a singualrity at a definite point f(x), on the sphere.