Stable Filtering Schemes with Rational Dilations

The relationship between multiresolution analysis and filtering schemes is a well known facet of wavelet theory. However, in the case of rational dilation factors, the wavelet literature is somewhat lacking in its treatment of this relationship. This work seeks to establish a means for the construction of stable filtering schemes with rational dilations through the theory of shift-invariant spaces. In particular, principal shift-invariant spaces will be shown to offer frame wavelet decompositions for rational dilations even when the associated scaling function is not refinable. Moreover, it will be shown that such decompositions give rise to stable filtering schemes with finitely supported filters, reminiscent of those studied by Kovacevic and Vetterli.