Abstract-Given a frame in C n which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashin's representation whose coefficients all have the smallest possible dynamic range O(1/ √ n). The information tends to spread evenly among these coefficients. As a consequence, Kashin's representations have a great power for reduction of errors in their coefficients, including coefficient losses and distortions.

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We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H n . Let h = (H 0 , H 1 , . . . , H n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices ⊆ R 2 such that the Gabor system {e 2πiλ 2 t h(t − λ 1 ) : λ = (λ 1 , λ 2 ) ∈ } is a frame for L 2 (R, C n+1 ). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ -function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames. Mathematics Subject Classification (2000)42C15 · 33C90 · 94A12

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We study problems of sampling and interpolation in a wide class of weighted spaces of entire functions. These weights are characterized by the property that their natural regularization as the envelop of the unit ball of the corresponding space is equivalent to the original weight. We give an independent description of such weights and also show that, in a sense, this is the widest class of weights and associated spaces for which results on sets of uniqueness, sampling, and interpolation related to the classical Paley-Wiener spaces can be extended in a direct and natural way, keeping the basic features of the theory intact. One of the basic tools for our study is the De Brange theory of spaces of entire functions.