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,
contrastingEast 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.
General
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