Legendre Polynomials and Their Use for Karhunen-Loève Expansion

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-Loève 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.