Feichtinger, Hans G.;Fornasier, M.

Flexible Gabor-wavelet atomic decompositions for ${{L}_2}$ Sobolev spaces

Annali di Matematica Pura e Applicata Vol.185 No.1 (2006) p.105--131

abstract

In this paper we present a general construction of frames, which allows to ensure that certain families of functions (atoms) obtained by a suitable combination of translation, modulation and dilation form Banach frames for the family of $L^2$-Sobolev spaces of any order. In this construction a parameter $alpha in [0,1)$ governs the dependence of the dilation factor % as a function of on the frequency parameter. The well-known Gabor and wavelet frames (also valid for the same scale of Hilbert spaces) using suitable Schwartz functions as building blocks arise as special cases ($alpha = 0$) and limiting case ($ alpha to 1)$ respectively. In contrast to those limiting cases it is no longer possible to use group theoretic arguments. Nevertheless we will show how to {it constructively} ensure that for Schwartz analysing atoms and any sufficiently dense, but discrete and well structured family of parameters one can guarantee the frame property. As a consequence of this novel constructive technique, one can generate {it quasi-coherent} dual frames by an iterative algorithm. As will be shown in a subsequent paper the newly frames introduced here generate Banach frames for corresponding families of $alpha$-modulation spaces.
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