Customer Reviews

Theory of Approximation of Functions of a Real Variable

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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.

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 reprint goes into classical analysis in a metric space. And specifically, the treatment often discusses a Hilbert space. This is important to theoretical physicists, as quantum mechanics is best understood with the pervasive usage of such spaces.

Anyway, Achieser deals with how to approximate a given function. Naturally, Fourier analysis figures prominently. Along with how to estimate the error when an infinite Fourier series is truncated.

The only pity is that someone has not authored a revision to this book, that takes into account maths developments since the 50s. Especially concerning fractals and wavelets. As these can now be used as basis functions in approximations, instead of the classical harmonic functions in Fourier analysis.