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Remarks about wavelet transforms and representations of groups

University of British Columbia

Gabor and wavelet transforms play an important role in signal and harmonic analysis. They are effective tools for localised time-frequency analysis. The goal of this paper is to look at the common features and at the differences between the two transforms. The Gabor and wavelet transforms are related to representations of groups: the representation of the Weyl-Heisenberg group and the representation of the affine group respectively, both acting on L²(R). The two transforms are also related to the Feichtinger-Grochenig theory, which generalises the notion o f frames to Banach spaces. Higher-dimensional analogues of the theory of continuous transforms in L²(R) are also of interest. Certain Lie groups obtained as the direct product of a closed subgroup of Rn with Rn, act on Rn such that the resulting representation is square-integrable. The theorem of Duflo-Moore gives the insight in understanding the synthesis o f elements in terms of the representation considered. An aspect of the differences between Gabor and wavelet transforms is the invertibility of the two frame operators. The Gabor frame operator is invertible on modulation spaces provided that it is invertible on L²(K), whereas the invertibility of the affine frame operator can depend on the function space.

Thesis/Dissertation

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2009-04-30

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http://hdl.handle.net/2429/7779

Mathematics

University of British Columbia

UBCV

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