The convergence problems of eigenfunction expansions of elliptic differential operators

In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harm...

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Main Author:	Ahmedov, Anvarjon A.
Format:	Conference or Workshop Item
Language:	English
Published:	IOP Publishing 2018
Subjects:	Q Science (General)
Online Access:	http://umpir.ump.edu.my/id/eprint/23294/ http://umpir.ump.edu.my/id/eprint/23294/ http://umpir.ump.edu.my/id/eprint/23294/1/The%20convergence%20problems%20of%20eigenfunction%20expansions%20of%20elliptic%20differential%20operators.pdf

id	ump-23294
recordtype	eprints
spelling	ump-232942019-05-16T06:27:39Z http://umpir.ump.edu.my/id/eprint/23294/ The convergence problems of eigenfunction expansions of elliptic differential operators Ahmedov, Anvarjon A. Q Science (General) In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained. The methods of approximation by multiple Fourier series summed over elliptic curves are applied to obtain suitable estimations for the maximal operator of the spectral decompositions. Obtaining of such estimations involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. In the present work the maximal operator of the elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville. IOP Publishing 2018-03 Conference or Workshop Item PeerReviewed pdf en http://umpir.ump.edu.my/id/eprint/23294/1/The%20convergence%20problems%20of%20eigenfunction%20expansions%20of%20elliptic%20differential%20operators.pdf Ahmedov, Anvarjon A. (2018) The convergence problems of eigenfunction expansions of elliptic differential operators. In: 3rd International Conference on Mathematics: Pure, Applied and Computation (ICoMPAC 2017), 1 November 2017 , Wyndham HotelSurabaya, Indonesia. pp. 1-9., 974 (1). ISSN 1742-6596 https://iopscience.iop.org/article/10.1088/1742-6596/974/1/012072/pdf
repository_type	Digital Repository
institution_category	Local University
institution	Universiti Malaysia Pahang
building	UMP Institutional Repository
collection	Online Access
language	English
topic	Q Science (General)
spellingShingle	Q Science (General) Ahmedov, Anvarjon A. The convergence problems of eigenfunction expansions of elliptic differential operators
description	In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained. The methods of approximation by multiple Fourier series summed over elliptic curves are applied to obtain suitable estimations for the maximal operator of the spectral decompositions. Obtaining of such estimations involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. In the present work the maximal operator of the elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville.
format	Conference or Workshop Item
author	Ahmedov, Anvarjon A.
author_facet	Ahmedov, Anvarjon A.
author_sort	Ahmedov, Anvarjon A.
title	The convergence problems of eigenfunction expansions of elliptic differential operators
title_short	The convergence problems of eigenfunction expansions of elliptic differential operators
title_full	The convergence problems of eigenfunction expansions of elliptic differential operators
title_fullStr	The convergence problems of eigenfunction expansions of elliptic differential operators
title_full_unstemmed	The convergence problems of eigenfunction expansions of elliptic differential operators
title_sort	convergence problems of eigenfunction expansions of elliptic differential operators
publisher	IOP Publishing
publishDate	2018
url	http://umpir.ump.edu.my/id/eprint/23294/ http://umpir.ump.edu.my/id/eprint/23294/ http://umpir.ump.edu.my/id/eprint/23294/1/The%20convergence%20problems%20of%20eigenfunction%20expansions%20of%20elliptic%20differential%20operators.pdf
first_indexed	2023-09-18T22:34:49Z
last_indexed	2023-09-18T22:34:49Z
_version_	1777416524853673984

The convergence problems of eigenfunction expansions of elliptic differential operators

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