Selected Topics in Asimptotic Geomwtric Analysis and Approximation Theory

Abstract

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.