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Selected Topics in Asimptotic Geomwtric Analysis and Approximation Theory Open Access


Other title
sinc function
Khinchine's type inequality
Compressive Sensing
Ball's integral inequality
Random Matrices
Prolate Spheroidal Wave Function
Type of item
Degree grantor
University of Alberta
Author or creator
Susanna, Spektor
Supervisor and department
Alexander, Litvak (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Bin, Han (Department of Mathematical and Statistical Sciences)
Vlad, Yaskin (Department of Mathematical and Statistical Sciences)
Nicole Tomczak-Jaegermann, (Department of Mathematical and Statistical Sciences)
Alexander, Litvak (Department of Mathematical and Statistical Sciences)
Alexander Penin (Department of Physics )
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
This thesis is mostly based on six papers on selected topics in Asymptotic Geometric Analysis, Wavelet Analysis and Applied Fourier Analysis. The first two papers are devoted to Ball's integral inequality. We prove this inequality via spline functions. We also provide a method for computing all terms in the asymptotic expansion of the integral in Ball's inequality, and indicate how to derive an asymptotically sharp form of a generalized Ball's integral inequality. The third paper deals with a Khinchine type inequality for weakly dependent random variables. We prove the Khinchine inequality under the assumption that the sum of the Rademacher random variables is zero. We also discuss other approaches to the problem. In particular, one may use simple random walks on graph, concentration and the chaining argument. As a special case of Khinchine's type inequality, we provide a tail estimate for a random variable with hypergeometric distribution, improving previously known estimates. The fourth paper devoted to the quantitative version of a Silverstein's Theorem on the 4-th moment condition for convergence in probability of the norm of a random matrix. More precisely, we show that for a random matrix with i.i.d. entries, satisfying certain natural conditions, its norm cannot be small. The fifth paper deals with Bernstein's type inequalities and estimation of wavelet coefficients. We establish Bernstein's inequality associated with wavelets. We also prove an asymptotically sharp form of Bernstein's type inequality for splines. We study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. We provide comparison of these two families. The sixth paper is on prolate spheroidal function. We prove that a function that is almost time and band limited is well represented by a certain truncation of its expansion in the Hermite basis.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
R.~Kerman and S.~Spektor. An asymptotically sharp form of Ball's integral inequality. arXive:1208.3799v1.R.~Kerman, R.~Ol'hava and S.~Spektor. An asymptotically sharp form of Ball's integral inequality. Proceedings of the AMS. July 15, 2013. PROC 130715.S.~Spektor. Khinchine inequality for Slightly dependent random variables. Proceedings of the AMS. November 2, 2013. PROC 131103.A.E.~Litvak and S.~Spektor. Quantitative version of a Silverstein's result. GAFA, Lecture Notes in Mathematics, Springer, Berlin. November 2, 2013.S.~Spektor and X.~Zhuang. Asymptotic Bernstein type inequalities and estimation of wavelets coefficients. Methods and Applications of Analysis, Vol. 19, No. 3 (2012), 289-312.

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