Differentiable positive definite kernels and Lipschitz continuity

Reade[11] has shown that positive definite kernels K ( x , t ) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O (1/ n 1+α ). In this paper we extend this result to positive definite kernels whose symmetric derivative K rr ( x , t ) = ∂ 2 r K ( x , t )/∂ x τ ∂ t τ is in Lip α and establish λ n ( K ) = O (1/ n 2 r +1+α ). If ∂ K rr /∂ t is in Lip α , the anticipated asymptotic estimate is also derived. The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.