On wavelet-based sampling Kantorovich operators and their study in multi-resolution analysis

TL;DR

This paper develops wavelet-based sampling operators using multiresolution analysis, providing approximation theorems, convergence rates, and empirical validation for their effectiveness in signal processing.

Contribution

It introduces a novel wavelet-based filtering framework with theoretical convergence analysis and practical examples, advancing multiresolution sampling methods.

Findings

Established a fundamental approximation theorem using wavelet transforms.

Derived convergence rates for wavelet-based filtering operators.

Validated theoretical results with empirical examples.

Abstract

In this work, wavelet-based filtering operators are constructed by introducing a basic function $D(t_1, t_2, t_3)$ using a general wavelet transform. The cardinal orthogonal scaling functions (COSF) provide an idea to derive the standard sampling theorem in multiresolution spaces which motivates us to study wavelet approximation analysis. With the help of modulus of continuity, we establish a fundamental theorem of approximation. Moreover, we unfold some other aspects in the form of an upper bound of the estimation taken between the operators and functions with various conditions. In that order, a rate of convergence corresponding to the wavelet-based filtering operators is derived, by which we are able to draw some important interferences regarding the error near the sharp edges and smooth areas of the function. Eventually, some examples are demonstrated and empirically proven to justify the fact about the rate of convergence. Besides that, some derivation of inequalities with justifications through examples and important remarks emphasizes the depth and significance of our work.