Eigenvalues of Analytic Kernels

Abstract

It is shown that the eigenvalues of an analytic kernel on a finite interval go to zero at least as fast as $R^{ - n} $ for some fixed $R < 1$. The best possible value of R is related to the domain of analyticity of the kernel. The method is to apply the Weyl–Courant minimax principle to the tail of the Chebyshev expansion for the kernel. An example involving Legendre polynomials is given for which R is critical.