Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations

TL;DR

This paper introduces efficient algorithms for computing connection coefficients between orthogonal polynomial families with polynomial or rational measure modifications, enabling fast spectral methods and matrix computations.

Contribution

It presents novel linear-complexity algorithms based on infinite-dimensional banded matrix factorizations for measure modifications of orthogonal polynomials.

Findings

Connection coefficients can be computed efficiently in linear time.

Modified classical weights support sparse differentiation matrices.

Algorithms enable fast computation of modified Jacobi matrices.

Abstract

We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.