Positive-Definiteness, Integral Equations and Fourier Transforms

Abstract

We show that positive definite kernel functions k(x, y), if continuous and integrable along the main diagonal, coincide with kernels of positive integral operators in L 2 (R).Such an operator is shown to be compact; under the further assumption k(x, x) → 0 as |x| → ∞ it is also trace class and the corresponding bilinear series converges absolutely and uniformly.If k 1/2 (x, x) ∈ L 1 (R), all these results are carried through to a 'rotated' Fourier transform: k(ν 1 , -ν 2 ) is the kernel of a compact positive operator and is represented by the absolutely and uniformly convergent series of Fourier transforms of eigenfunctions.The trace of the operator is an invariant under Fourier transforms.