From filters to wavelets via direct limits

Details From filters to wavelets via direct limits From filters to wavelets via direct limits

2006-02-21

arxiv.org/abs/math/0602450v1

Mathematics

Functional Analysis

6 pages, to appear in GPOTS 2005 proceedings

Scientific paper

We 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.

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