Two Banach spaces of atoms for stable wavelet frame expansions

Abstract

It is well known that for sufficiently nice wavelet functions (e.g., Schwartz functions with a few vanishing moments) the regularity of the wavelet transform allows to recover any L 2 -function in a stable way 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 2 ( R d ) . In the present paper a systematic treatment of mild sufficient conditions for the validity of such a statement is provided on the basis of two new Banach spaces of functions, to be denoted by F 0 ( R d ) and F 1 ( R d ) . Their norms 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 a new system, based using sufficiently close irregular set (in the sense of small jitter error), and using sufficiently small modification of the atom (in terms of one of the two norms), is an irregular wavelet frame of similar quality. Whereas, it is shown that the perturbation may occur in the sense that every parameter is allowed to be perturbed by the same amount for atoms in F 0 ( R d ) with arbitrary time-scale sequences, one is allowed to modify wavelet frames for atoms from the strictly larger class F 1 ( R d ) in a similar way if the sampling pattern forms an affine lattice (similar to classical wavelet systems).