Robustness of Regular Sampling in Sobolev Algebras
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It is the purpose of this paper to feature the link between the theory of minimal norm interpolation over lattices by elements fromSobolev algebras H s (S d )with what is known as the theory of spline-type (or principal shift invariant) spaces. As an extremely useful tool allowing us to establish various kinds of robustness results we will present the Wiener amalgam spacesW(B, l p with general (smooth) local components and globall p behavior. For this reason a summary of their most important properties, including convolution relations and the behavior under the Fourier transform, is presented.
The discussion of projection and minimal norm interpolation operators is not restricted to the pure Hilbert space setting for which these concepts were developed originally. Among others we show LP-stability of the (for p = 2 orthogonal) projection from LP onto the corresponding spline-type spaces with lP-coefficients.
As a main result (which can be formulated in several different concrete ways) we show that for s > d/2 the mapping f ↦ Qs, a(f), from f to the minimal norm interpolation in Hs over the lattice aℤd, a >, depends continuously on the input parameters (s, a). It also extends to certain fractional LP-Sobolev spaces consisting of continuous functions in LP. In this modified setting the outcome of this procedure depends continuously on (s, a) in the LP sense. Moreover, the mapping is robust against small jitter errors. Wiener amalgam spaces tum out to be very useful, both for a precise formulation and in the proofs of such results.
KeywordsBanach Space Reproduce Kernel Hilbert Space Riesz Basis Atomic Decomposition Closed Linear Span
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