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MiS Preprint

Power Series Kernels

Barbara Zwicknagl


We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as used in Machine Learning, certain new nonlinearly factorizable kernels and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain native Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case.

Dec 20, 2006
Dec 20, 2006
MSC Codes:
41A05, 41A10, 41A25, 41A58, 41A63, 68T05
multivaraite polynomial approximation, bernstein theorem, dot product kernels, reproducing kernel Hilbert spaces, error bounds, convergence orders

Related publications

2009 Journal Open Access
Barbara Zwicknagl

Power series kernels

In: Constructive approximation, 29 (2009) 1, pp. 61-84