Fast associated classical orthogonal polynomial transforms

TL;DR

This paper introduces a fast approximate method for converting associated classical orthogonal polynomials to classical ones, leveraging eigenvalue problem linearization and discrete transforms for efficient computation.

Contribution

It presents a novel approach that formulates the problem as a quadratic eigenvalue problem and extends divide-and-conquer algorithms for efficient polynomial transform computation.

Findings

Quadratic eigenvalue problem formulation for associated orthogonal polynomials

Extension of divide-and-conquer approach to block upper-triangular banded matrices

Efficient synthesis of singular integral transforms using discrete sine and cosine transforms

Abstract

We discuss a fast approximate solution to the associated classical -- classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue problem with polynomial coefficients such that the differential operator is degree-preserving. Upon linearization, the discretization of this quadratic eigenvalue problem is block upper-triangular and banded. After a perfect shuffle, we extend a divide-and-conquer approach to the upper-triangular and banded generalized eigenvalue problem to the blocked case, which may be accelerated by one of a few different algorithms. Associated orthogonal polynomials arise from iterated Stieltjes transforms of orthogonal polynomials; hence, fast approximate conversion to classical cases combined with fast discrete sine and cosine transforms provides a modular mechanism for synthesis of singular integral transforms of classical orthogonal polynomial expansions.