Extremal Polynomials and Riemann Surfaces

by Andrei Bogatyrev, Nikolai Kruzhilin (Translator)

Overview

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.​

Product Details

ISBN-13:	9783642443329
Publisher:	Springer Berlin Heidelberg
Publication date:	06/11/2014
Series:	Springer Monographs in Mathematics
Edition description:	2012
Pages:	150
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

About the Author

The author is working in the field of complex analysis, Riemann surfaces and moduli, optimization of numerical algorithms, mathematical physics. He was awarded the S.Kowalewski Prize in 2009 by the Russian Academy of Sciences

Table of Contents

1 Least deviation problems.- 2 Chebyshev representation of polynomials.- 3 Representations for the moduli space.- 4 Cell decomposition of the moduli space.- 5 Abel’s equations.- 6 Computations in moduli spaces.- 7 The problem of the optimal stability polynomial.- Conclusion.- References.

Editorial 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)

From the Publisher