Two dimensional pseudo Chebyshev wavelets and their application in the theory of approximation.

Abstract

For the first time in 2022, authors introduced the notion of pseudo-Chebyshev wavelets in one- dimensional. Continuing the study in an advanced sense, in this article, two-dimensional pseudo-Chebyshev wavelets are introduced. Two dimensional pseudo Chebyshev wavelet expansion of a function of two variables is defined and verified. This research paper introduces a novel algorithm based on the two-dimensional pseudo-Chebyshev wavelet method to address computation problems in approximation theory. The methods are illustrated by an example and compared with prominent Chebyshev wavelet methods to demonstrate the validity and applicability of the results. The error analysis and convergence analysis of a function $\xi$  in the  H\"{o}lder classes have been studied by these wavelets. Moreover, the error of approximation of functions of Holder's class has been estimated by orthogonal projection operators of its two-dimensional pseudo-Chebyshev wavelets. The results of this paper are the significant developments in wavelet analysis.