Wavelets and graph $C^*$-algebras

TL;DR

This paper surveys the connection between wavelet theory and graph $C^*$-algebras, introduces new generalizations of wavelets for higher-rank graphs, and explores potential applications in network traffic analysis.

Contribution

It presents novel methods for constructing wavelets of arbitrary shapes and extends spectral graph wavelets to higher-rank graphs, broadening the theoretical framework.

Findings

Introduction of cubical wavelets for higher-rank graphs

Construction of wavelets of arbitrary shapes

Extension of spectral graph wavelets to higher-rank graphs

Abstract

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.