Extremal Polynomials and Riemann Surfaces

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.

List 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.

Report

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)

Product details

• Authors: Andrei Bogatyrev
• Assisted by: Nikolai Kruzhilin (Translation)
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 07.06.2012
• EAN: 9783642256332
• ISBN: 978-3-642-25633-2
• No. of pages: 150
• Weight: 408 g
• Illustrations: XXVI, 150 p.
• Series: Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; B, Physics, Mathematics and Statistics, Mathematical and Computational Engineering, Theoretical, Mathematical and Computational Physics, Mathematical physics, Numerical analysis, Manifolds (Mathematics), Differential calculus & equations, Engineering mathematics, Applied mathematics, Numerical and Computational Physics, Simulation, Mathematical and Computational Engineering Applications, Maths for engineers, Functions of a Complex Variable, Functions of complex variables, Global analysis (Mathematics), Global Analysis and Analysis on Manifolds, Approximations and Expansions, Approximation theory