Theory of Approximation of Functions of a Real Variable (ISSN) by A. F. Timan (PDF)

Description

Theory of Approximation of Functions of a Real Variable discusses a number of fundamental parts of the modern theory of approximation of functions of a real variable. The material is grouped around the problem of the connection between the best approximation of functions to their structural properties.This text is composed of eight chapters that highlight the relationship between the various structural properties of real functions and the character of possible approximations to them by polynomials and other functions of simple construction. Each chapter concludes with a section containing various problems and theorems, which supplement the main text. The first chapters tackle the Weierstrass’s theorem, the best approximation by polynomials on a finite segment, and some compact classes of functions and their structural properties. The subsequent chapters describe some properties of algebraic polynomials and transcendental integral functions of exponential type, as well as the direct theorems of the constructive theory of functions. These topics are followed by discussions of differential and constructive characteristics of converse theorems. The final chapters explore other theorems connecting the best approximations functions with their structural properties. These chapters also deal with the linear processes of approximation of functions by polynomials. The book is intended for post-graduate students and for mathematical students taking advanced courses, as well as to workers in the field of the theory of functions.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Even at a time when a subject undergoes a revolution, and new directions keep coming, new methods; even brand new and different applications— even then are there central ideas that are constant, the fundamentals. Within mathematics, approximation theory is such a field: In the past decade, we have seen a host of such new developments: wavelet approximations, fast computational algorithms with applications to turbulence, chaos and fractals; computational efficiencies from scaling similarities, and data compression; and new adaptive non-linear algorithms. The author Achieser of this Dover classic (first published by Ungar in 1956, and reprinted by Dover in 1992) is a pioneer in the approximation theory, and the book is still a very attractive first. I recommend it to students even today! Achieser’s lovely book begins with the fundamentals on normed spaces, and it has the classical theorems of Weierstrass, Muntz, and Riesz. The other topics range from the incisive ideas of Tchebysheff and Haar to harmonic approximation. From extremal properties of transcendental functions to Wiener’s general Tauber theorem, and more.Review by Palle Jorgensen, October 2003.

⭐An excellent, graduate level, reference book on approximation theory.Different aspects of relations between constructive and structural properties of functions studied.

⭐This should be on the reading list of every graduate student in control or signal processing. This book is an encyclopedia of results in approximation theory including Chebyshev approximation, harmonic analysis, and extremal properties of integral transcendental functions. The exposition is terse in some places and the proofs are sometimes sketchy, but the examples are really great. The focus on the ideas is excellent.

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