标题： 勒让德多项式及其在Karhunen-Loève展开中的应用 显示英文标题

标题： Legendre Polynomials and Their Use for Karhunen-Loève Expansion

摘要： 本文做出了两个主要贡献。 首先，我们提供了一个教学性的回顾，推导了Legendre多项式的三项递推关系，而无需依赖经典的Legendre微分方程、Rodrigues公式或生成函数。 这个阐述旨在对本科生易于理解。 其次，我们开发了一个计算框架，用于超矩形域上各向同性高斯随机场的Karhunen-Loève展开。 该框架利用Legendre多项式及其相关的高斯求积，并且在更高空间维度下仍保持高效。 首先通过理论上有依据的初始化进行牛顿优化拟合，将协方差核近似为平方指数的非负混合。 所得的可分离核使得Legendre-Galerkin离散化以单个维度的Kronecker乘积形式呈现，子矩阵表现出偶/奇对称结构。 在组装过程中，我们引入了一种Duffy型变换，随后进行求积。 这些结构性质相比直接方法显著减少了内存使用和算术成本。 所有算法和数值实验都提供在一个开源仓库中，可以重现本工作中每一个图表。 显示英文摘要

摘要： 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.