Extremal Polynomials and Riemann Surfaces

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Springer Science & Business Media, May 31, 2012 - Mathematics - 150 pages

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.

 

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Contents

Chapter 1 Least Deviation Problems
1
Chapter 2 Chebyshev Representation of Polynomials
14
Chapter 3 Representations for the Moduli Space
29
Chapter 4 Cell Decomposition of the Moduli Space
53
Chapter 5 Abels Equations
73
Chapter 6 Computations in Moduli Spaces
89
Chapter 7 The Problem of the Optimal Stability Polynomial
115
Conclusion
135
References
137
Further Reading
144
Index
147
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About the author (2012)

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

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