Eigenvalue distribution of Mercer‐like kernels

Abstract

Abstract We study eigenvalues of positive definite kernels of L 2 integral operators on arbitrary intervals. Assuming integrability and uniform continuity of the kernel on the diagonal, we show that the eigenvalue distribution is totally determined by the smoothness of the kernel together with its decay rate at infinity along the diagonal. Moreover, the rate of decay of eigenvalues depends on both these quantities in a symmetrical way. Our main result treats all possible orders of differentiability and all possible rates of decay of the kernel; the known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)