Function spaces obeying a time-varying bandlimit

TL;DR

This paper introduces a new class of function spaces that model signals with a time-varying bandwidth, providing a mathematical framework for adaptive signal processing techniques.

Contribution

It establishes the equivalence of these spaces to existing models and offers a rigorous way to define and utilize local bandwidth in signal analysis.

Findings

Provides a mathematical model for time-varying bandwidth

Shows equivalence to Hardy space and Herglotz measure models

Enables construction of adaptive signal processing methods

Abstract

Motivated by applications to signal processing and mathematical physics, recent work on the concept of time-varying bandwidth has produced a class of function spaces which generalize the Paley-Wiener spaces of bandlimited functions: any regular simple symmetric linear transformation with deficiency indices $(1,1)$ is naturally represented as multiplication by the independent variable in one of these spaces. We explicitly demonstrate the equivalence of this model for such linear transformations to several other functional models based on the theories of meromorphic model spaces of Hardy space and purely atomic Herglotz measures on the real line, respectively. This theory provides a precise notion of a time-varying or local bandwidth, and we describe how it may be applied to construct signal processing techniques that are adapted to signals obeying a time-varying bandlimit.