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The problem of approximating a given quantity is one of the oldest challenges faced by mathematicians. Its increasing importance in contemporary mathematics has created an entirely new area known as Approximation Theory. The modern theory was initially developed along two divergent schools of thought: the Eastern or Russian group, employing almost exclusively algebraic methods, was headed by Chebyshev together with his coterie at the Saint Petersburg Mathematical School, while the Western mathematicians, adopting a more analytical approach, included Weierstrass, Hilbert, Klein, and others.…mehr

Produktbeschreibung
The problem of approximating a given quantity is one of the oldest challenges faced by mathematicians. Its increasing importance in contemporary mathematics has created an entirely new area known as Approximation Theory. The modern theory was initially developed along two divergent schools of thought: the Eastern or Russian group, employing almost exclusively algebraic methods, was headed by Chebyshev together with his coterie at the Saint Petersburg Mathematical School, while the Western mathematicians, adopting a more analytical approach, included Weierstrass, Hilbert, Klein, and others. This work traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century contributions of Sergei Bernstein in defining a new branch of function theory. One of the key strengths of this book is the narrative itself. The author combines a mathematical analysis of the subject with an engaging discussion of the differing philosophical underpinnings in approach as demonstrated by the various mathematicians. This exciting exposition integrates history, philosophy, and mathematics. While demonstrating excellent technical control of the underlying mathematics, the work is focused on essential results for the development of the theory. The exposition begins with a history of the forerunners of modern approximation theory, i.e., Euler, Laplace, and Fourier. The treatment then shifts to Chebyshev, his overall philosophy of mathematics, and the Saint Petersburg Mathematical School, stressing in particular the roles played by Zolotarev and the Markov brothers. A philosophical dialectic then unfolds, contrasting East vs. West, detailing the work of Weierstrass as well as that of the Goettingen school led by Hilbert and Klein. The final chapter emphasizes the important work of the Russian Jewish mathematician Sergei Bernstein, whose constructive proof of the Weierstrass theorem and extension of Chebyshev's work serve to unify East and West in their approaches to approximation theory. Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation. TOC:Dedication.- Foreword.- Preface.- Introduction.- Forerunners.- Pafnuti Lvovich Chebyshev.- The Saint Petersburg Mathematical School.- Development Outside Russia.- Constructive Function Theory: Kharkiv.- Biographies.- Explanations.- References.- Index.- List of Figures.- List of Tables.

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Autorenporträt
Karl-Georg Steffens, Universität Duisburg-Essen, Duisburg, Germany
Inhaltsangabe
Dedication
Foreword
Preface
Introduction
Forerunners
1.1 Euler s Analysis of Delisle s Map
1.2 Laplace s Approximation of Earth s Surface
Pafnuti Lvovich Chebyshev
2.1 Chebyshev s Curriculum Vitae
2.2 Stimuli for the Development of a Theory
2.3 First Theoretical Approaches
2.4 First Theoretical Compositions
2.5 Theory of Orthogonal Polynomials
2.6 Other Contributions of P. L. Chebyshev
2.7 Chebyshev - Euler of the 18th Century?
The Saint Petersburg Mathematical School
3.1 Aleksandr Nikolaevich Korkin
3.2 Egor Ivanovich Zolotarev
3.3 Andrey and Vladimir Andreevich Markov
3.4 Julian Karol Sochocki
3.5 Konstantin Aleksandrovich Posse
3.6 A. A. Markov s Lectures
3.7 Résumé
Development Outside Russia
4.1 The Mediator: Felix Klein
4.2 Blichfeldt s Note
4.3 Kirchberger s Thesis
4.4 Other Non-Quantitative Contributions
4.5 On Convergence and Series Expansions
4.6 Fejér and Runge
4.7 Quantitative Approximation Theory
4.8 Jackson s Thesis
4.9 A Note About Göttingen s Role
Constructive Function Theory: Kharkiv
5.1 Antony-Bonifatsi Pavlovich Psheborski
5.2 A Short Biography of Sergey Natanovich Bernstein
5.3 First Contributions to Approximation Theory
5.4 Constructive Function Theory as the Development of Chebyshev s Ideas
Biographies...
A.1 Matvey Aleksandrovich Tikhomandritski
A.2 Nikolaj Yakovlevich Sonin
A.3 Aleksandr Vasilevich Vasilev
A.4 Ivan Lvovich Ptashitski
A.5 Dmitry Fedorovich Selivanov
A.6 Aleksandr Mikhaylovich Lyapunov
A.7 Ivan Ivanovich Ivanov
A.8 Dmitry Alksandrovich Grave
A.9 Georgi Feodosievich Voronoy
Explanations
B.1 Russian Academic Degrees
References
Index
List of Figures
List of Tables
Rezensionen
Aus den Rezensionen:

"... Ich möchte dieses kleine Werk nicht mehr missen, denn Karl-Georg Steffens hat ein echtes Kleinod zur frühen Geschichte der Approximationstheorie vorgelegt und seine Begeisterung für die Leistungen der russischen Schule ist ehrlich und begründet! ... Das Werk ist meines Wissens nach einzigartig in der Literatur zur Geschichte der Approximationstheorie. ... Der Text ist flüssig geschrieben und gut lesbar ... Ich wünsche dem Buch zahlreiche Leser und kann dem Verlag nur raten, seine ... Werbung für diese Publikation zu verstärken. Hier liegt ein Schatz!" -- Thomas Sonar, in: Mathematische Semesterberichte, 2007, Vol. 54, S. 122 f.