Extending wavelet filters. Infinite dimensions, the non-rational case, and indefinite-inner product spaces

TL;DR

This paper broadens the theoretical framework of wavelet filters by exploring infinite-dimensional, non-rational cases using indefinite inner product spaces, connecting with Krein spaces, generalized Schur functions, and Cuntz relations.

Contribution

It introduces a novel approach to wavelet filters incorporating indefinite inner product spaces and extends the analysis to non-rational cases, linking with advanced functional analysis tools.

Findings

Established connections between non-rational wavelet filters and Krein spaces.

Linked generalized Schur functions with wavelet filter stability.

Extended wavelet filter theory to infinite-dimensional and indefinite inner product spaces.

Abstract

In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the non-rational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations.