Approximation Methods by Singular Integral Operators
Date of Award
Dissertation (Access Restricted)
Doctor of Philosophy
George A Anastassiou
Ebenezer Olusegun George
In this dissertation I study the basic approximation properties of the general Picard, Gauss-Weierstrass and Poisson-Cauchy singular integral operators over the real line, which are not positive linear operators. In particular I study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. This is given via inequalities and with the use of higher order modulus of smoothness of the high order derivative of the involved function. Some of these inequalities are proved to be attained. Also I study the global smoothness preservation property of these operators. Furthermore, I give asymptotic expansions, Voronovskaya type, of the error of approximation. At the end, I study related properties of the general fractional Gauss-Weierstrass singular integral operators. There properties are study with respect to Lp norm, 1<=p<=infinity. For the convenience of the reader, the chapters of this dissertation are written in a self-contained style. Chapter 1 provides extensive motivation and applications regarding my presented research.
Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Mezei, Razvan Alexandru, "Approximation Methods by Singular Integral Operators" (2011). Electronic Theses and Dissertations. 2222.