Compactly supported, orthogonal, continuous piecewise polynomial multiresolution analysis

TL;DR

The paper addresses constructing compactly supported, orthogonal MRAs for piecewise polynomial splines with $C^0$ continuity by developing ramp-based generators expressible through Jacobi polynomials and deriving explicit hypergeometric representations for scaling functions. It provides closed-form Mellin and Fourier transforms of these functions, introduces new MRAs with rational coefficients, and outlines a practical algorithm for computing refinement coefficients and associated wavelets. The work uniquely links special function theory, notably Jacobi and hypergeometric polynomials, with multiresolution analysis to yield structurally rich, spectrally analyzable MRAs suitable for fixed-point arithmetic and signal processing applications. These contributions advance explicit constructions of $C^0$ orthogonal MRAs, offering precise analytical tools for spectral analysis and potential numerical advantages in practice.

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.

Theorems & Definitions (33)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Theorem 5
• proof
• Lemma 6
• proof