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Extremal Polynomials and Riemann Surfaces [electronic resource] /by Andrei Bogatyrev.

by Bogatyrev, Andrei [author.]; SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Springer Monographs in Mathematics: Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : 2012.Description: XXV, 150 p. 47 illus. online resource.ISBN: 9783642256349.Subject(s): Mathematics | Functions of complex variables | Global analysis | Numerical analysis | Engineering mathematics | Mathematics | Functions of a Complex Variable | Approximations and Expansions | Numerical Analysis | Global Analysis and Analysis on Manifolds | Numerical and Computational Physics | Appl.Mathematics/Computational Methods of EngineeringDDC classification: 515.9 Online resources: Click here to access online
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.
In: Springer eBooksSummary: 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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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.

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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