Harmonic Analysis On Affine Groups: Generalized Continuous Wavelet Transforms
Date
2021-08-18T14:30:28Z
Authors
Milad, Raja
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Abstract
The set of all invertible affine transformations of a two dimensional real vector space forms a locally compact group G2 that is isomorphic to the semi-direct product group formed when GL2(R) acts on R2 in the obvious manner, where GL2(R) denotes the group of 2 by 2 real matrices with nonzero determinant. We give an explicit decomposition of the left regular representation of G2 as a direct sum of infinitely many copies of a single irreducible representation. We also obtain an analog of the continuous wavelet transform associated to the representation we identify.
Description
We introduced the background that needed latter for the thesis. We study Peter-Weyl theorem for compact groups and apply it for specific non-trivial group. We gave the full details about the Duflo-Moore operator, and the continuous wavelet transform associated with the square integrable representation of the group. In chapter 4, the factorization theorem that we discovered and prove it by two different ways. Then we used this theorem in chapter 5 to give the full details about the project group. Finding the Duflo-Moore operator, and the continuous wavelet transform associated with the square integrable representation of the group. The wavelet that we got is a novel wavelet transform. This will analyze functions in three dimensions. This transform will have great affect if implemented in machine software like Matlap. We think will add more features to image compression analysis.
Keywords
Affine Groups, Continuous Wavelet Transforms, Harmonic analysis, Generalized continuous Wavelet Transforms