Fast Conversion Algorithms for Orthogonal Polynomials

TL;DR

This paper presents efficient algorithms for converting between orthogonal polynomial bases and the monomial basis, including a new algorithm for the inverse conversion, both maintaining the same computational complexity.

Contribution

The paper introduces a novel, complexity-preserving algorithm for converting monomials back to orthogonal polynomial bases, complementing existing conversion methods.

Findings

The algorithms operate with the same complexity for both conversions.

A new inverse conversion algorithm is proposed and analyzed.

The methods improve efficiency in polynomial basis transformations.

Abstract

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation.