On Alpert Multiwavelets

TL;DR

This paper explores the multiresolution analysis of Alpert multiwavelets, providing explicit formulas for matrix coefficients using hypergeometric functions and analyzing their solutions to eigenvalue and difference equations.

Contribution

It introduces explicit formulas for Alpert multiwavelet matrix coefficients and examines conditions for unique solutions in wavelet equations.

Findings

Explicit formulas for matrix coefficients in terms of hypergeometric functions

Solutions to generalized eigenvalue and difference equations

Conditions for uniqueness of wavelet matrix solutions

Abstract

The multiresolution analysis of Alpert is considered. Explicit formulas for the entries in the matrix coefficients of the refinement equation are given in terms of hypergeometric functions. These entries are shown to solve generalized eigenvalue equations as well as partial difference equations. The matrix coefficients in the wavelet equation are also considered and conditions are given to obtain a unique solution.