Approximation in Hilbert spaces of the Gaussian and other weighted power series kernels
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This article considers linear approximation based on function evaluations in reproducing kernel Hilbert spaces of the Gaussian kernel and a more general class of weighted power series kernels on the interval [-1, 1]. We derive almost matching upper and lower bounds on the worst-case error, measured both in the uniform and L^2([-1,1])-norm, in these spaces. The results show that if the power series kernel expansion coefficients α_n^-1 decay at least factorially, their rate of decay controls that of the worst-case error. Specifically, (i) the nth minimal error decays as α_n^ -1/2 up to a sub-exponential factor and (ii) for any n sampling points in [-1, 1] there exists a linear algorithm whose error decays as α_n^ -1/2 up to an exponential factor. For the Gaussian kernel the dominating factor in the bounds is (n!)^-1/2.
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