Study on wavelets constructed from rational filters

Abstract

In the recent, the compactly supported wavelets, whose filters have finite impulse response, have been studied by Chui, Daubechies, Vetteli and many others in mathematics and signal processing. In this thesis, we study the infinitely supported wavelets constructed from the rational filters, which have the exponential decay. We present the wavelet frames with the rational filters from the Butterworth-type filters and the $Battle-Lemarié$ -type filters using the Unitary Extension Principle of Ron and Shen. In addition, we show that the asymptotic limit of the generators is the Shannon wavelet. We present the polynomial dual filters for the rational filters and give the algorithm to find the duals.