2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/109045We present a new proof of a theorem of Mallat which describes a construction of wavelets starting from a quadrature mirror filter. Our main innovation is to show how the scaling function associated to the filter can be used to identify a certain direct limit of Hilbert spaces with $L^2(\R)$ in such a way that one can immediately identify the wavelet basis. Our arguments also use a pair of isometries introduced by Bratteli and Jorgensen, and exploit the geometry inherent in the Cuntz relations satisfied by these isometries.6 pages, to appear in GPOTS 2005 proceedingsFunctional AnalysisOperator Algebras42C40text