Parseval frames from compressions of Cuntz algebras

• Published in: Mathematische Zeitschrift Vol. 304; no. 1
• Main Authors: Christoffersen, Nicholas, Dutkay, Dorin Ervin, Picioroaga, Gabriel, Weber, Eric S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2023; Springer Nature B.V
• Subjects: Hilbert space; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Random walk; Vectors (mathematics); Fractal measures; 42C10; Walsh bases; Cuntz algebra; Iterated function systems; Row co-isometry; 28A80; Parseval frame; 05C81; Fourier series; 42A16; 47L55
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-023-03259-w

A row co-isometry is a family ( V i ) i = 0 N - 1 of operators on a Hilbert space, subject to the relation ∑ i = 0 N - 1 V i V i ∗ = I . As shown in Bratteli et al. (J Oper Theory, 43, 97–143, 2000), 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.