Legendre Polynomials and Their Use for Karhunen-Lo\`eve Expansion

TL;DR

This paper provides an accessible review of Legendre polynomials and introduces an efficient computational framework for Karhunen-Lo extbackslash e extquotesingle ve expansions of isotropic Gaussian fields using Legendre polynomials, suitable for higher dimensions.

Contribution

It offers a pedagogical derivation of Legendre polynomials and develops a scalable, efficient method for Karhunen-Lo extbackslash e extquotesingle ve expansions leveraging Legendre-Galerkin discretization.

Findings

Efficient algorithms for high-dimensional Gaussian field expansions.

Structural properties reduce memory and computational costs.

Open-source implementation reproduces all results.

Abstract

This paper makes two main contributions. First, we present a pedagogical review of the derivation of the three-term recurrence relation for Legendre polynomials, without relying on the classical Legendre differential equation, Rodrigues' formula, or generating functions. This exposition is designed to be accessible to undergraduate students. Second, we develop a computational framework for Karhunen-Lo\`eve expansions of isotropic Gaussian random fields on hyper-rectangular domains. The framework leverages Legendre polynomials and their associated Gaussian quadrature, and it remains efficient even in higher spatial dimensions. A covariance kernel is first approximated by a non-negative mixture of squared-exponentials, obtained via a Newton-optimized fit with a theoretically informed initialization. The resulting separable kernel enables a Legendre-Galerkin discretization in the form of a Kronecker product over single dimensions, with submatrices that exhibit even/odd parity structure. For assembly, we introduce a Duffy-type transformation followed by quadrature. These structural properties significantly reduce both memory usage and arithmetic cost compared to naive approaches. All algorithms and numerical experiments are provided in an open-source repository that reproduces every figure and table in this work.