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Compactly Supported Wavelets and Representations of the Cuntz Relations

✍ Scribed by Ola Bratteli; David E Evans; Palle E.T Jorgensen

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
284 KB
Volume
8
Category
Article
ISSN
1063-5203

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✦ Synopsis

We study the harmonic analysis of the quadrature mirror filters coming from multiresolution wavelet analysis of compactly supported wavelets. It is known that those of these wavelets that come from third order polynomials are parameterized by the circle, and we compute that the corresponding filters generate irreducible mutually disjoint representations of the Cuntz algebra O 2 except at two points on the circle. One of the two exceptional points corresponds to the Haar wavelet and the other is the unique point on the circle where the father function defines a tight frame which is not an orthonormal basis. At these two points the representation decomposes into two and three mutually disjoint irreducible representations, respectively, and the two representations at the Haar point are each unitarily equivalent to one of the three representations at the other singular point.

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