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Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials
1st Edition - February 10, 2022
Authors: Robert B. Gardner, Narendra K. Govil, Gradimir V. Milovanović
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Inequalities for polynomials and their derivatives are very important in many areas of mathematics, as well as in other computational and applied sciences; in particular they play… Read more
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Inequalities for polynomials and their derivatives are very important in many areas of mathematics, as well as in other computational and applied sciences; in particular they play a fundamental role in approximation theory. Here, not only Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, but also ones for trigonometric polynomials and related functions, are treated in an integrated and comprehensive style in different metrics, both on general classes of polynomials and on important restrictive classes of polynomials. Primarily for graduate and PhD students, this book is useful for any researchers exploring problems which require derivative estimates. It is particularly useful for those studying inverse problems in approximation theory.
Applies Markov-Bernstein-type inequalities to any problem where derivative estimates are necessary
Presents complex math in a clean and simple way, progressing readers from polynomials into rational functions, and entire functions of exponential type
Contains exhaustive references with more than five hundred citations to articles and books
Features methods to solve inverse problems across approximation theory
Includes open problems for further research
Graduate and PhD students working in mathematical analysis and approximation theory, especially in geometry of polynomials and complex approximation
Table of Contents
About the authors
Chapter 1: History and introduction: classical Markov–Bernstein inequalities
1.1. Markov's theorem and some of its generalizations and refinements
1.2. Markov–Duffin–Schaeffer inequalities and generalizations
1.3. Bernstein's inequality and some of its generalizations and refinements
1.4. Bernstein inequalities on several intervals and Szegő variants
Chapter 2: Different types of Bernstein inequalities
2.1. Bernstein-type inequalities concerning growth of polynomials
2.2. Bernstein-type inequalities for rational functions
2.3. Bernstein-type inequalities in the Lr norm
2.4. Bernstein-type inequalities in other environments
2.5. Bernstein-type inequalities for entire functions of exponential type
Chapter 3: Extremal problems of Markov–Bernstein type in integral norms
3.1. Orthogonal polynomials and classical weight functions
3.2. Extremal problems of Markov type in L2 norm for the classical weights
3.3. Different modifications of weighted L2 Markov–Bernstein's extremal problems and inequalities
3.4. Extremal problems of Markov type in Lr norm
3.5. Extremal problems of Markov type on the restricted classes of polynomials
Chapter 4: Bernstein-type inequalities for polynomials with restricted zeros
4.1. Inequalities for polynomials with zeros outside of a disk
4.2. Inequalities for polynomials with zeros inside of a disk
4.3. Inequalities for self-inversive and self-reciprocal polynomials
4.4. Inequalities for some other classes of polynomials
Chapter 5: Bernstein-type inequalities in the Lr norm
5.2. Bernstein's inequality in Lr
5.3. The case of r>0
Chapter 6: Bernstein-type inequalities for polar derivatives of polynomials
6.2. Extensions of the classical theorems of Grace and Laguerre
6.3. Bounds on the uniform norm of polar derivative of a polynomial
6.4. Bounds on the integral mean values of the polar derivative of a polynomial
No. of pages: 442
Published: February 10, 2022
Paperback ISBN: 9780128119884
eBook ISBN: 9780128120071
Robert B. Gardner
Robert Gardner is a tenured Professor of Mathematics and Statistics at East Tennessee State University specializing in Bernstein-type inequalities. He has co-authored/co-edited two books, including Real Analysis with an Introduction to Wavelets and Applications that was published by Elsevier in 2005.
Affiliations and expertise
Professor of Mathematics and Statistics, Department of Mathematics and Statistics, East Tennessee State University, TN, USA
Narendra K. Govil
Narendra K. Govil is Professor Emeritus in the Department of Mathematics and Statistics at Auburn University, from where he retired as Professor, in 2020. He received his M.Sc. from Aligarh Muslim University, India and Ph.D. from the University of Montreal, Canada. He is a Fellow of the National Academy of Sciences, India and has been Alumni Professor in Department of Mathematics and Statistics at Auburn University. Before joining Auburn in 1983, he was a Professor at Indian Institute of Technology (IIT), New Delhi, India. He is a researcher in Complex Analysis and Approximation Theory, and has written a large number of papers in subjects related to Bernstein-type Inequalities, Geometry of the Zeros of Polynomials, Special Functions, and Wavelets. He is presently serving as Editor/Associate Editor of several journals, and has co-authored/co-edited six books including Progress in Approximation Theory and Applicable Complex Analysis, published by Springer in 2017.
Affiliations and expertise
Professor Emeritus in the Department of Mathematics and Statistics, Auburn University, AL, USA
Gradimir V. Milovanović
Gradimir V. Milovanović is a Professor of Numerical Analysis and Approximation Theory and Full Member of the Serbian Academy of Sciences and Arts. He studied at University of Niš, obtaining a B.Sc. (1971) in electrical engineering and computer sciences and an M.Sc. (1974) and a Ph.D. (1976) in mathematics.
He was with the Faculty of Electronic Engineering and the Department of Mathematics at the same place as, promoted to professor (1986) and acting as Dean of the Faculty of Electronic Engineering (2002-2004) and Rector of the University of Niš (2004–06), as well as Dean of the Faculty of Computer Sciences at the Megatrend University, Belgrade (2008-2011), until he joined the Mathematical Institute of the Serbian Academy of Sciences and Arts in Belgrade (2011). He was President of the National Council for Scientific and Technological Development of the Republic of Serbia (2006-2010).
His research interests are Orthogonal Polynomials and Systems; Interpolation, Quadrature Processes and Integral Equations; Approximation by Polynomials and Splines; Extremal Problems, Inequalities and Zeros of Polynomials. He published 7 monographs, about 250 scientific papers in refereed journals, 35 book chapters, about 50 papers in conference proceedings, as well as 20 textbooks. Most significant monograph works of Milovanović are Topics in Polynomials: Extremal Problems, Inequalities, Zeros (coauthors: D. S. Mitrinović and Th. M. Rassias), published at over 800 pages by World Scientific (Singapore, 1994) and known in the world as „Bible of Polynomials“ and the monograph Interpolation Processes – Basic Theory and Applications (cоаuthor: G. Mastroianni) by Springer, 2008. (Home page: http://www.mi.sanu.ac.rs/~gvm/ ). He is currently serving as an Editor-in-Chief and an Associate Editor for several journals (Journal of Inequalities and Applications, Springer; Optimization Letters, Springer; Applied Mathematics and Computation, Elsevier; Publications de l’Institut Mathématique, Mathematical Institute, Belgrade, etc.).
Affiliations and expertise
Professor of Numerical Analysis and Approximation Theory and Full Member, Serbian Academy of Sciences and Arts, Beograd, Serbia