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Compactly supported wavelets and representations of the Cuntz relations

Bratteli, Ola, Evans, David Emrys and Jorgensen, Palle E.T 2000. Compactly supported wavelets and representations of the Cuntz relations. Applied and Computational Harmonic Analysis 8 (2) , pp. 166-196. 10.1006/acha.2000.0283

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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 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: wavelet ; Cuntz algebra ; representation ; orthogonal expansion ; quadrature mirror filter ; isometry in Hilbert space
ISSN: 1063-5203
Last Modified: 04 Jun 2017 04:08

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