Overview
- Includes numerous problems and exercises which provide a deep insight in the subject and allow to conduct independent research in this topic
- Contains many pictures which visualize involved theory
- Description of effective computational algorithms for higher genus algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (7 chapters)
Keywords
About this book
The problems of conditional optimization of the uniform (or C-) norm for polynomials and rational functions arise in various branches of science and technology. Their numerical solution is notoriously difficult in case of high degree functions. The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmüller theory, foliations, braids, topology are applied to approximation problems.
The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.
Reviews
From the reviews:
“This book develops the classical Chebyshev approach to optimization problems in polynomial spaces. This approach yields an analytical representation for the solution in terms of Riemann surfaces. The text includes numerous problems, exercises, and illustrations. … In this book, methods from various areas of mathematics are used. … It has more than 150 pages throughout which the author makes a lot of effort to give as many results as possible, and yet provide lots of details to make the reading easier.” (Konstantin Malyutin, Zentralblatt MATH, Vol. 1252, 2012)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Extremal Polynomials and Riemann Surfaces
Authors: Andrei Bogatyrev
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-642-25634-9
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2012
Hardcover ISBN: 978-3-642-25633-2Published: 07 June 2012
Softcover ISBN: 978-3-642-44332-9Published: 11 June 2014
eBook ISBN: 978-3-642-25634-9Published: 31 May 2012
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XXVI, 150
Additional Information: Original Russian edition published by MCCME, Moscow, 2005
Topics: Functions of a Complex Variable, Approximations and Expansions, Numerical Analysis, Global Analysis and Analysis on Manifolds, Numerical and Computational Physics, Simulation, Mathematical and Computational Engineering