Robustness of Sampling in Sobolev Algebras

It is the purpose of this paper to feature the link between the theory of minimal norm interpolation over lattices by elements from Sobolev algebras Hs(R) with what is known as the theory of spline-type (or principal shift invariant) spaces. As an extremely useful tool allowing to establish various kinds of robustness results we will present the so-called Wiener Amalgam spaces W (B, `), with general (smooth) local components and global ` 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 L-stability of the (for p = 2 orthogonal) projection from L onto the corresponding spline-type spaces with `-coefficients. As a main result (which can be formulated in several different concrete ways) we show that for s > d/2 the mapping f 7→ Qs,a(f), from f to the minimal norm interpolation in Hs over the lattice aZd, a > 0, depends continuously on the input parameters (s, a). It also extends to certain fractional L-Sobolev spaces consisting of continuous functions in L. In this modified setting the outcome of this procedure depends continuously on (s, a) in the L-sense. Moreover, the mapping is robust against small jitter errors. Wiener amalgam spaces turn out to be very useful, both for a precise formulation and in the proofs of such results. 1The first name author wants to thank the Dept. of Applied Mathematics at the University of Heidelberg for hospitality during the time when this paper was finished. The work of the second named author was supported by the EU project NetAGES, IST-1999-29034. 2 Hans G. Feichtinger and Tobias Werther