Interpolatory filter banks and interpolatory wavelet packets

Abstract

In classic wavelet sampling, research is generally focused on interpolatory bases in individual spaces. Consideration is seldom given to relations among different interpolatory bases, including those potentially related by filter banks. In this paper, we show that when an interpolatory basis exists in a space with a multiresolution analysis, this space can be decomposed orthogonally into two subspaces in which interpolatory bases also exist. Furthermore, similarly to standard orthogonal bases, these interpolatory bases can be related by pairs of filters, called interpolatory filter banks. For such interpolatory filter banks, it is shown that existence of interpolatory scaling functions generally leads to that of interpolatory wavelets and interpolatory wavelet packets. With this result, we further propose new algorithms for constructing such interpolatory wavelets/wavelet packets from corresponding scaling functions. In examples, our algorithms are applied to some typical wavelet spaces, demonstrating our theorems for interpolatory wavelets, interpolatory wavelet packets and interpolatory filter banks.