On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method

In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by comparing the Riesz and Cesaro methods of summation for eigenfunction expansions of the Laplace-Beltrami operator. It is shown that the weak localization principle for the integrable functions f(x) at the point x depends not only on behavior of the function around x but on the behavior of the function around diametrically opposite point \overline{x}

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

Summation by riesz means of the fourier-laplace series

In this work, weinvestigate conditions for summability of the Fourier-Laplace series of integrable functions by Riesz means. The kernel of Riesz means is estimated through comparison with the Cesaro means. Properties of D and D* points are required in obtaining this estimation. © 2018, University of Nis. All rights reserved

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

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 uniform summability of the Fourier-Laplace series on the sphere

Convergence problems has been the focus of interest for researchers that are working in the fields of spectral theory. In the current research we investigate issues relating to the summability of the Fourier-Laplace series on the unit sphere. The necessary conditions which are required to obtain good estimation for summability of the Fourier-Laplace series investigated. This research will also provide new and sufficient conditions in the form of theorems and lemmas which will validate the uniform summability of the Fourier-Laplace series on the sphere

Solving Nonstiff Higher Order Odes Using Variable Order Step Size Backward Difference Directly

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis

Numerical integration of functions from holder classes Hs [0, 1] by linear legendre multi wavelets

In the previous research, a direct computational method based on linear Legendre multi-wavelets has been applied for solving definite integrals. However, the error analysis to show the convergence of the method has not been discussed. Therefore, error analysis of the approximation method is established in the Holder classes Hs[0, 1] to show the efficiency of the method. The connections of the module of difference smoothness of the function is also established. Finally, some numerical examples of the implementation the method for the functions from Holder classes are presented

Asymptotic formula for the Riesz means of the spectral functions of Laplace-Beltrami operator on unit sphere

The mathematical models of the heat and mass transfer processes on the ball type solids can be solved using the theory of convergence of Fourier-Laplace series on unit sphere. Many interesting models have divergent Fourier-Laplace series, which can be made convergent by introducing Riesz and Cesaro means of the series. Partial sums of the Fourier-Laplace series summed by Riesz method are integral operators with the kernel known as Riesz means of the spectral function. In order to obtain the convergence results for the partial sums by Riesz means we need to know an asymptotic behavior of the latter kernel. In this work the estimations for Riesz means of spectral function of Laplace-Beltrami operator which guarantees the convergence of the Fourier-Laplace series by Riesz method are obtained