arXiv math/06024502006-02-21math.FAmath.OA

From filters to wavelets via direct limits

Nadia S. Larsen, Iain Raeburn

备注：: 6 pages, to appear in GPOTS 2005 proceedings

英文摘要：

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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