Generalized Haar Wavelets and Frames

Generalized Haar wavelets were introduced in connection with the problem of detecting specific periodic components in noisy signals. John Benedetto and I showed that the non–normalized continuous wavelet transform of a periodic function taken with respect to a generalized Haar wavelet is periodic in time as well as in scale, and that generalized Haar wavelets are the only bounded functions with this property. In this paper, I shall discuss generalized Haar wavelets in a discrete setting. I shall present a characterization of all generalized Haar wavelets which have the property that our discretized version of the continuous wavelet transform is a topological isomorphism onto its range. This is equivalent to the fact that the set of analysis vectors used constitute a frame for l(Z). A similar result is obtained for l(Z). Generalized Haar wavelets allow a fast computation of a discretized version of the continuous wavelet transform of a function, as I shall show. I shall present examples of generalized Haar wavelets and calculate the corresponding frame bounds and analysis filter banks.