On Modified Mellin-Gauss-Weierstrass Convolution Operators

Abstract

Mellin transform has various applications to real-life problems in function approximation, signal processing, and image recognition, thus, it has been the main ingredient of many studies in diverse fields. This study is devoted to Mellin operators and their variants to improve approximation accuracy and approximate ratio. Two Mellin type operators are reconstructed by using two sequences of functions to enable lower pointwise approximation error as well as higher pointwise convergence rate. Keeping the idea of Mellin convolution, these classes aim to be associated with functions defined on the semi-real axis, and the affine and quadratic functions pairs are fixed points. It has been shown, both theoretically and numerically, that operators can be used to approximate functions pointwise. Indeed the approximation accuracy can be adjusted by tuning the parameters. Moreover, weighted approximation, as well as Voronovskaya type results, are studied throughout the paper. The advantages of each operator over the other in terms of both approximation errors and convergence rates are presented.