Convergence of Frame Series from Hilbert spaces to Banach spaces And l^1-boundedness
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Yu, Pu-Ting
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Abstract
This thesis consists of four parts. In the first part of the thesis, we study the convergence of frame expansions from separable Hilbert spaces to separable Banach spaces, and modulation spaces. A complete characterization of Schauder frames for which the associated frame expansions converge unconditionally for every alternative and every element will be presented.
The second part is devoted to the conjecture proposed by Aldroubi et al. regarding the frame property of sequences generated by normal operators. By introducing a new notion named frame-normalizability, we provide several partial answers to this conjecture.
We will focus on new notions named \ell^1-boundedness and \ell^1-frame-boundedness in the third part, where we consider the certain generalizations of Fourier coefficients from L^2(T) to separable Hilbert spaces. Instead of focusing on the exponential family we will work with bases that are topologically isomorphic to orthonormal bases and even frames. Two questions along with direction are of our main interest. First, under what circumstances is \ell^1-boundedness equivalent to \ell^1-frame-boundedness? Second, is the collection of \ell^1-bounded sets closed under sums and unions? For the first question, we will prove that several cases are true. For the second question, we will present several intriguing implications following from the assumptions that the collection of \ell^1-bounded sets closed under sums and unions.
Finally, we will present some frames with special structures. For example, we will construct Gabor frames with atoms with poor time-frequency decay.
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2024-07-22
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Dissertation