A Class of non-MRA Band-limited Wavelets

Abstract

We give a characterization of a class of band-limited wavelets of $L^2({\mathbb R})$ and show that none of these wavelets come from a multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$ of ${\mathbb R}$ which is symmetric with respect to the origin. We give necessary and sufficient conditions on a function $\psi\in L^2({\mathbb R})$ with supp $\hat\psi\subseteq S_n$ to be an orthonormal wavelet. This result generalizes the characterization of a class of wavelets of E. Hern\'andez and G. Weiss. The dimension functions associated with these wavelets are also computed explicitly. Starting from the wavelets we have constructed, we are able to construct examples of wavelets in each of the equivalence classes of wavelets defined by E. Weber.