Differentiable positive definite kernels and Lipschitz continuity
Cochran, J.A. and Lukas, M.A. (1988) Differentiable positive definite kernels and Lipschitz continuity. Mathematical Proceedings of the Cambridge Philosophical Society, 104 (02). p. 361.
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Reade[ll] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order a on a bounded region have eigenvalues which are asymptotically O(l/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) ≡ ∆2rK(x, t)/∆xr∆tr is in Lipα and establish λn(K) = O(l/n2r+1+α). If ∆Krr/∆t is in Lipα, the anticipated asymptotic estimate is also derived. The proofs use a well-known result of Chang , recently rederived by Ha , 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.
|Publication Type:||Journal Article|
|Murdoch Affiliation:||School of Mathematical and Physical Sciences|
|Publisher:||Cambridge University Press|
|Copyright:||© 1988, Cambridge Philosophical Society.|
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