Parseval Frames from Compressions of Cuntz Algebras

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}