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In digital signal processing it is often advantageous to analyze a given signal using an adaptive method. The signal is approximated or represented as a superposition of "basic" waveforms chosen from a <i>dictionary</i> of such waveforms so as to best match the signal. The matching pursuit algorithm of Mallat and Zhang is such a method and is discussed in… (More)

We study the question of lower bounds for the Hausdorff dimension of a set in R n containing spheres of every radius. If n ≥ 3 then such a set must have dimension n. If n = 2 then it must have dimension at least 11/6. We also study the analogous maximal function problem and related problem of Besicovitch sets with an axis of symmetry. Besicovitch and Rado… (More)

In this paper we consider smooth analogues of operators studied in connection with pointwise convergence of the solution of the free Schrödinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we obtain sharp L 2 → L 2 bounds. We begin this paper by giving… (More)

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