Optimal Recovery in Translation { invariant Spaces of Functions
نویسنده
چکیده
Starting from optimal recovery (in the sense of Micchelli, Rivlin, and Winograd) of functions in reproducing kernel Hilbert spaces from function values at scattered data points, we show that any continuously embedded translation{invariant Hilbert subspace H of L 2 (IR d ) that allows continuous point evaluation is necessarily principal, i.e. it is the native space of a positive de nite function . If, in addition, H is invariant under orthogonal transformations of IR d , the function necessarily is a positive de nite radial basis function. This serves to show that positive de nite radial basis functions and their native spaces arise very naturally and are by no means exotic theoretical constructions.
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