Compactly supported, orthogonal, continuous piecewise polynomial multiresolution analysis

TL;DR

This paper provides explicit hypergeometric function representations for orthogonal multiresolution analysis scaling functions, introduces new analyses with rational coefficients, and derives their Mellin and Fourier transforms.

Contribution

It offers explicit formulas and new multiresolution analyses with rational coefficients, expanding the mathematical tools for wavelet analysis.

Findings

Explicit hypergeometric representations of scaling functions.

Closed-form Mellin and Fourier transforms for these functions.

Introduction of new multiresolution analyses with rational coefficients.

Abstract

We present explicit representations in terms of hypergeometric functions for the scaling functions in the $C^0$ orthogonal multiresolution analyses associated with piecewise continuous polynomials. Closed formulas for the Mellin transform of these functions as well as their Fourier transforms are derived. Some new multiresolution analyses whose scaling functions have coefficients that are rational numbers are introduced and discussed.