Parseval frames of piecewise constant functions

TL;DR

This paper introduces a novel method to construct Parseval frames of piecewise constant functions in L^2[0,1], utilizing operators related to Cuntz relations, and demonstrates how these frames can be extended to orthonormal bases.

Contribution

It presents a new construction of Parseval frames for piecewise constant functions using operators satisfying Cuntz-type relations without isometry, and shows their dilation to orthonormal bases.

Findings

Constructed Parseval frames similar to generalized Walsh bases

Operators satisfy Cuntz-type relations without being isometries

Frames can be dilated to orthonormal bases

Abstract

We present a way to construct Parseval frames of piecewise constant functions for $L^2[0,1]$. The construction is similar to the generalized Walsh bases. It is based on iteration of operators that satisfy a Cuntz-type relation, but without the isometry property. We also show how the Parseval frame can be dilated to an orthonormal basis and the operators can be dilated to true Cuntz isometries.