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Title: Operators in complex analysis and the affine group
Author: El Mabrok, Abdelhamid Salem A.
ISNI:       0000 0004 2745 6921
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2012
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This thesis is devoted to providing a detailed description and construction of intertwining operators related to L2-type spaces in terms of representations of the affine group Aff. We review numerous connections between unitary operators, provide decompositions of the wavelet transforms from the affine group. The group naturally acts by the quasi- regular representation on the space L2 (R) of square integrable functions on the real line. The Hardy space H2(R) is an irreducible invariant subspace under such an action. A eo- adjoint representation of this group spatially splits into irreducible components supported on the orbits, which turns out to be half-lines and {0}. The intertwining operator between quasi-regular and co-adjoint representations turns out to be the Fourier transform. This provides a background for wavelets technique in the theory of complex operators. We analyze the construction and origin of unitary operators describing the structure of the space of continuous wavelet transforms inside the space of all square integrable function on Aff with the left Haar measuer dv- from the viewpoint of induced representations. We show that these operators are intertwining operators among pairs of induced representations of the affine group Aft. A characterization of the space of wavelet transforms using the Cauchy-Riemann type equations is given.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available