Two Banach spaces of atoms for stable wavelet frame expansions

In principle it is well known that for sufficiently nice wavelet functions the regularity of the wavelet transform allows to recover any L-function from its samples over any sufficiently dense, irregular sampling set. Equivalently, the (irregular) set of affine transforms of the given wavelet function forms a frame for L(R). In the present paper a systematic treatment of sufficient conditions for the validity of such statement is provided, on the basis of two new Banach spaces of functions, denoted by F0 and F1 in the sequel, using classical concepts (e.g. norms involving derivatives etc.). The norms on these spaces also turn out to be highly suitable for the description of perturbation results. Given an irregular wavelet frame using an atom from one of these spaces implies that for any sufficiently close irregular set (in the sense of small jitter error), and sufficiently small modification of the atom (in terms of one of the two norms). Whereas it is shown that the perturbation may occur in the sense that every parameter is allowed to be perturbed in the same size for atoms in F0 with arbitrary time-scale sequences, one is allowed to modify wavelet frames for atoms from the strictly larger class F1 in a similar way if the sampling pattern forms an affine lattices (similar to classical wavelet systems). ? This work was supported partially by the FWF project P-15605 of the Austrian Science Foundation, the K.C.Wong Education Foundation, the National Natural Science Foundation of China (10201014 and 60472042), the Program for New Century Excellent Talents in University, and the Research Fund for the Doctoral Program of Higher Education. ∗ Correspoding author Email addresses: [email protected] (Hans G. Feichtinger), [email protected] (Wenchang Sun), [email protected] (Xingwei Zhou). Preprint submitted to Elsevier Science 7 October 2005