Parseval Frames from Compressions of Cuntz Algebras

• Published in: arXiv.org
• Main Authors: Christoffersen, Nicholas, Dorin Ervin Dutkay, Picioroaga, Gabriel, Weber, Eric
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.01.2023
• Subjects: Hilbert space; Mathematical analysis; Operators (mathematics); Random walk; Vectors (mathematics)
• ISSN: 2331-8422

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}