We introduce three families of positive definite bivariate integral kernels whose eigenfunctions and eigenvalues can be explicitly derived in terms of classical orthogonal polynomials. All sequences of classical orthogonal polynomials are involved in at least one of our developments. We show that the well-known Karhunen–Loève expansions of the Wiener, Brownian bridge and Anderson–Darling Gaussian stochastic processes are particular cases of our results.

中文翻译：

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与经典正交多项式相关的三个 Karhunen-Loève 展开族

我们介绍了三族正定二元积分核，其特征函数和特征值可以根据经典正交多项式明确导出。经典正交多项式的所有序列至少涉及我们的一项发展。我们表明，众所周知的维纳、布朗桥和安德森-达令高斯随机过程的 Karhunen-Loève 展开是我们结果的特殊情况。