Eigenvalues of smooth kernels

Extract

Suppose is a symmetric square integrable kernel on the unit square [0, 1]². Then

is a compact symmetric operator on the Hilbert space L²[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvalues

of T satisfy as n → ∞. We prove a related result. Let W¹²[0, 1]² denote the space of all K(x, t) ε L²[0, 1]² which are absolutely continuous in x for each t and absolutely continuous in t for each x, and the partial derivatives ∂K/∂x(x, t), ∂K/∂t(x, t) are both in L²[0, 1]². We slow that the eigenvalues of any satisfy .