Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials

TL;DR

This paper establishes a connection between Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials, providing explicit formulas and algorithms for their computation using classical polynomial techniques.

Contribution

It introduces explicit formulas for Legendre-Angelesco polynomials and Alpert multiwavelets, and develops algorithms for their computation via Cholesky factorization and Jacobi matrices.

Findings

Explicit formulas for Legendre-Angelesco polynomials

Algorithms for computing multiwavelets using Cholesky factorization

Method for calculating matrices in the scaling relation for various multiplicities

Abstract

We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the multiwavelets.