The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back to the days of the discovery of the Periodic table by the Russian Chemist Dmitri Mendeleev. In Chapter 2, we narrow down the contents stated in Chapter 1 to the problems we were interested in working during the course of this dissertation. Henceforth, we present a problem formulation mainly for those results for which solutions or partial solutions are provided in the subsequent chapters. Over the years Bernstein inequality has been generalized and extended in several directions. In Chapter 3, we establish rational analogues to some Bernstein-type inequalities for restricted zeros and prescribed poles. Our inequalities extend the results for polynomials, especially which are themselves improved versions of the classical Erdös-Lax and Turán inequalities. In working towards proving our results, we establish some auxiliary results, which may be of interest on their own. Chapters 4 and 5 focus on the research carried out with the Askey-Wilson operator applied on polynomials and entire functions (of exponential type) respectively. In Chapter 4, we first establish a Riesz-type interpolation formula on the interval [−1, 1] for the Askey-Wilson operator. In consequence, a sharp Bernstein inequality and a Markov inequality are obtained when differentiation is replaced by the Askey-Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein-type inequality in L2−space. We conclude this chapter with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail. Chapter 5 is devoted to an intriguing application of the Askey-Wilson operator. By applying it on the Sampling Theorem on entire functions of exponential type, we obtain a series representation formula, which is what we called an extended Boas’ formula. Its power in discovering interesting summation formulas, some known and some new will be demonstrated. As another application, we are able to obtain a couple of Bernstein-type inequalities. In the concluding chapter, we state some avenues where this research can progress.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Puwakgolle Gedara, Rajitha, "In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions" (2018). Electronic Theses and Dissertations, 2004-2019. 6077.