Parseval Frames from Compressions of Cuntz Algebras

TL;DR

This paper introduces new methods for constructing Parseval frames in Hilbert spaces using compressions of Cuntz algebra representations, with applications to Fourier and Walsh bases on specific measures and intervals.

Contribution

It presents novel constructions of Parseval frames via iterated operators and random walks on graphs, extending the theory of frames from Cuntz algebra representations.

Findings

Constructed Parseval Fourier bases on self-affine measures.

Developed Parseval Walsh bases on the interval.

Connected frame constructions to random walks on finite graphs.

Abstract

A row co-isometry is a family $(V_i)_{i=0}^{N-1}$ of operators on a Hilbert space, subject to the relation $$\sum_{i=0}^{N-1}V_iV_i^*=I.$$ As shown in \cite{BJK00}, row co-isometries appear as compressions of representations of Cuntz algebras. In this paper we will present some general constructions of Parseval frames for Hilbert spaces, obtained by iterating the operators $V_i$ on a finite set of vectors. The constructions are based on random walks on finite graphs. As applications of our constructions we obtain Parseval Fourier bases on self-affine measures and Parseval Walsh bases on the interval. \end{abstract}