Integral Equation for Simultaneous Diagonalization of Two Covariance Kernels

Let Ki(s,t) and K2(s,t), --T^s,t^T,be real, symmetric, continuous and strictly positive-definite kernels, and denote by Kj and 883 884 T H E BELL SYSTEM T E C H N I C A L JOURNAL, M A Y - J U N E 1967 K2 the integral operators with kernels Ki(s,t) and K2(s,t). We have previously1 established that, if is a densely defined and bounded operator on £2 (the space of all square-integrable functions on [--T,T]) and if its extension to the whole of £2 has eigenvalues X, and complete orthonormal eigenfunctions