The History of Approximation Theory - From Euler to Bernstein

Exciting exposition integrates history, philosophy, and mathematics
Combines a mathematical analysis of approximation theory with an engaging discussion of the differing philosophical underpinnings behind its development
Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation

List of contents

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

Summary

* Exciting exposition integrates history, philosophy, and mathematics

* Combines a mathematical analysis of approximation theory with an engaging discussion of the differing philosophical underpinnings behind its development

* Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation

Additional text

From the reviews:

"This book aims to tell the historical evolution of the methods and results of approximation theory, starting from the work of Euler in 1777 on minimizing distance errors in maps of Russia, and of Laplace in 1843 on finding the best ellipsoid for the earth, and ending with the work of Bernstein. The central character in the book is Chebyshev (1821–1894) and the Saint Petersburg Mathematical School he created.... [The] book contains much interesting material and has certainly motivated me to learn more about the subject and its history. No book can wish for more and so I recommend the book to anyone interested in approximation theory or in the history of mathematics."        —MAA Reviews

"The author describes the early stage of development of approximation theory. … The famous Russian school founded by … Chebyshev is presented in full. … contributions of German and French mathematicians to both Weierstrass and Chebyshev directions are illuminated … . All this is supported with quotations from documents, letters, speeches, early publications, and memoirs. … Undoubtedly, the author has succeeded in catching the spirit both of the Russian atmosphere of that time and of the depth and richness of the human relationships … ." (B. Boyanov, Mathematical Reviews, Issue 2006 j)

“The discussion of mathematical results and ideas constitutes the main part of the book—always accompanied by moving stories, marked by devotion to science, by humanity, and often by the cruelty of the political turbulence of Russian history. Undoubtedly, the author has succeeded in catching the spirit both of the Russian atmosphere of that time and of the depth and richness of the human relationships of the people involved.”(MATHEMATICAL REVIEWS)

"This nice little book is devoted to the early development of approximation theory. It mainly concentrates on the role of Russianmathematicians but confronts it with the development of approximation theory in the West … The book will be of interest for historians of mathematics and for specialists in approximation theory." (EMS Newsletter, September, 2006)

"The storie is vivid, rich and touching. Many interesting details from the life, education and work of the unsurpassed master of classical analysis, Sergei Natanovich Bernstein, are given. Some of them are published for the first time and come from personal reminiscences of his pupils, close friends and collaboratiors. ...All this is supported with quotations from documents, letters speeches, easrly publications, and memiors. ... Undoubtedly, the author has succeeded in catching the spirit both of the Russian atmosphere of that time and of the depth and richness of the human relationships of the people involved." (American Mathematical Society)

"The book’s stated goal is ‘to describe the early development of approximation theory… (until) 1919….’ I learned lots of new names, new stories, and new connections. … In conclusion, this book is necessary … for studying the history of approximation theory. … it does contain plenty of useful information that is not readily or not at all available elsewhere." (Paul Nevai, SIAM Review, Vol. 49 (1), 2007)

Report

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.)

Product details

• Authors: K. Steffens, Karl-Georg Steffens
• Assisted by: Georg A Anastassiou (Editor), George A Anastassiou (Editor), George A. Anastassiou (Editor)
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2005
• EAN: 9780817643539
• ISBN: 978-0-8176-4353-9
• No. of pages: 219
• Weight: 350 g
• Illustrations: 12 SW-Abb., 2 Tabellen, XIX, 219 p. 22 illus.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; B, History, Mathematics, Mathematics and Statistics, Differential calculus & equations, Sequences, Series, Summability, Calculus & mathematical analysis, Sequences (Mathematics), History of Mathematical Sciences, Functional analysis & transforms, Fourier Analysis, Approximations and Expansions, Approximation theory