Some integral equations with 'nonrational' kernels

Abstract

Fredholm integral equations of the first and second kind arise in many problems in statistical communication theory. However, almost all cases in which solutions are known, for equations over finite intervals, involve covariance kernels with rational Fourier transforms. We present solutions for two classes of "non-rational" kernels--triangular kernels of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(t) = \max [0, 1 - |t|]</tex> and Gauss-Markov kernels of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(t, s) = f(t)g(s), t \leq s, R(t, s) = f(s)g(t), s \leq t</tex> . We also treat kernels that are combinations of the two types.