Power Series Kernels
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Submission date: 20. Dec. 2006 (revised version: October 2007)
published in: Constructive approximation, 29 (2009) 1, p. 61-84
DOI number (of the published article): 10.1007/s00365-008-9012-4
MSC-Numbers: 41A05, 41A10, 41A25, 41A58, 41A63, 68T05
Keywords and phrases: multivaraite polynomial approximation, bernstein theorem, dot product kernels, reproducing kernel Hilbert spaces, error bounds, convergence orders
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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.