| Abstract: | A wavelet is a bandpass filter with the additional time (or space) localization capability. The bandwidth and center frequency of this filter, as well as the width of its time (or space) localization-window, are adjusted by changing the values of the scaling (or dilation) parameter. Hence, in the study of multivariate (or multidimensional) phenomena, each time (or space) variable should possess its own scaling parameter in order to allow maximal flexibility in time-frequency and/or phase-space analysis. The notion of multifrequency wavelets, to be introduced in this writing, is based on this point of view. The objective of this chapter is to derive the Littlewood-Paley properties of these wavelets and their duals, to study the multivariate (and directional) multiresolution analysis and its approximation power, and to give an efficient multi-frequency wavelet decomposition algorithm. One of the advantages of this approach is the possibility of generating a dyadic Riesz basis of L^2 (R^s), s>=1, by using a single function. This is indeed the case, and we will demonstrate the special features of such Riesz bases by considering the tensor-product setting. |
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