Numerical Integration and Hilbert-Schmidt Norms

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Peter Zinterhof


We consider numerical integration procedures of the type \\frac1N\\sum f(x_a)\\bar g(x_a)\\to\\int f(x)\\bar g(x)\\,dx on Hilbert spaces with a reproducing kernel over the polish space E. It turns out that the Hilbert-Schmidt norm of certain operators play an important role as an estimator for the efficiency of the underlying integration procedure. Applications of these general results for the important cases of integration over the s-dimensional unit cube E=[0,1)s, and several numerical experiments will be shown. All methods discussed in the present paper are inherently parallel.

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