Characterization of Riesz bases of wavelets generated from multiresolution analysis |
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Authors: | Bin Han Rong-Qing Jia |
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Affiliation: | aDepartment of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 |
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Abstract: | We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a compact operator on a certain Banach space of sequences. With the help of the spectral theory of the compact operator we are able to characterize the convergence of the cascade algorithm associated with an exponentially decaying mask in terms of the spectrum of the corresponding transition operator. As an application of this study we establish the main result of this paper which gives a complete characterization of all possible Riesz bases of compactly supported wavelets generated from multiresolution analysis. Several interesting examples are provided to illustrate the general theory. |
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Keywords: | Riesz bases Wavelets Cascade operators Transition operators Compact operators Spectra Cascade algorithms |
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