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GABOR AND WAVELET FRAMES. GEOMETRY AND APPLICATIONS

GABOR AND WAVELET FRAMES. GEOMETRY AND APPLICATIONS,PIOTR WOJDY L LO

GABOR AND WAVELET FRAMES. GEOMETRY AND APPLICATIONS  
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The theory of frames was born in 1952 as a part of the non- harmonic analysis. However, its expansive growth in late 80's is due to the perspective of real-life applications. In the thesis we deal rst with the abstract theory of the orbits of a single vector under the action of unitary transformation group that are tight frames, named here coherent tight frames. The relations with the algebraic aspects of the problem are concerned. In particular, we use the representation theory to analyze phenomena observed recently by researchers in Gabor and wavelet analysis, in particular, berization technique. We nd a partial characteri- zation of vectors generating coherent tight frames. We apply the theory to obtain some new results for discrete subgroups of Heisenberg group. Under a certain technical assumption we give the representation formula for the frame operator in the discrete wavelet case. In the second part of the thesis we deal with the question how tight frames are related to orthonormal bases. We study closely a geometry of Gabor tight frames with bound 2 and this of Wilson basis constructed in 1992. The ge- ometrical characterization of the functions generating Wilson basis is given. We discuss this outcome confronted with P.G. Casazza's result about a repre- sentation of a frame by means of orthonormal bases. We complete with the simple argument that Wilson bases are unconditional in Banach spaces related to the group action. These spaces are known in the literature as coorbit or modulation spaces. We apply this approach to obtain results on the values of exact frame bounds for Gabor and wavelet systems. We illustrate theorems with a numer- ical presentation. We generalize also the Littlewood-Paley type inequalities introduced for the systems in 1993 by C. Chui and X. Shi. In the thesis also a new insight into the berization approach developed in 1995-1998 by A. Ron and Z. Shen is given.
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