On Generalized Walsh Bases

Abstract

This paper continues the study of orthonormal bases (ONB) of $L^{2}[0,1]$ introduced in Dutkay et al. (J. Math. Anal. Appl. 409(2):1128–1139, 2014) by means of Cuntz algebra $\mathcal{O}_{N}$ representations on $L^{2}[0,1]$ . For $N=2$ , one obtains the classic Walsh system. We show that the ONB property holds precisely because the $\mathcal{O}_{N}$ representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.