A treatise on trigonometric series. Vol. I, II. Authorized translation by Margaret F. Mullins. (English) Zbl 0129.28002

Bary, N. K.

Oxford-London-New York-Paris-Frankfurt: Pergamon Press. xxiii, 553 p.; xix, 508 p. (1964).

This is an English translation of the Russian original of 1961 (Moscow-Leningrad).
One does not need to recommend this excellent textbook for students who want to know more than a usual undergraduate course in trigonometric series. The book is readable for students and the author certainly does not misuse the words “easily seen”. The original was a one volume book of 936 pages; the English translation is printed very attractively in two volumes.
The translator states that the first volume (Chapters I–VI) is suitable for University students and the second one for advanced students and research workers. This is not obvious to the reviewer; the second volume seems to be neither more special nor more difficult than the first one, and both volumes form a whole.
The first one contains the introductory material, basic concepts and theorems, investigation of Fourier coefficients, convergence of a Fourier series at a point, Fourier series of continuous functions, and problems of convergence and divergence of a Fourier series in a set. In the second volume the author deals with summability, conjugate trigonometric series, absolute convergence of Fourier series, series with monotone coefficients, lacunary series, convergence, divergence and absolute convergence of general trigonometric series, problems of uniqueness, and representation of a function by a trigonometric series. Each volume contains Bibliography and Index.
There is one important difference between the original book and the translation, namely, the translation contains sets of Problems added at the end of each of the Chapters in Volume I by P. L. Ul’yanov. There are together 93 Problems mostly of theoretic type which increase the value of the, book.
However, the translation repeats some misprints which actually existed in the original and adds a number of more important errors which are mostly due to the translator’s work. The language of the translation happens to be mathematically incorrect. As an example I shall quote one of the Problems (Problem 5 on p. 205 Vol. I): “If functions \(g(x)\in C(0, 2\pi)\) and \(f(x)\in C(0, 2\pi)\) exist such that \(g(x) = f(x)\) for \(x\in [1, 2]\), the Fourier series of \(g(x)\) and \(f(x)\) are however not uniformly equiconvergent in the interval \((1, 2)\).”
There are even more serious defects. For instance, a wrong translation of the definition of a \(T^*\)-method on p. 11, Vol. I, produces an obvious contradiction with p. 236, Vol. II. Fatou’s lemma in the formulation on p. 26, Vol. I, is false. “Integral function” means exactly the same as “entire function”, and not a function with values \(1, 2, \ldots, n\) as suggests Lemma 2 on p. 366, Vol. II. There is no use to write the whole list of errors here in order to conclude that the book was worthwhile a more careful translation.

Reviewer: Julian Musielak (Poznań)

MSC:

42-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42Axx	Harmonic analysis in one variable

Keywords:

Fourier coefficients; convergence of Fourier series at a point; Fourier series of continuous functions; convergence and divergence of Fourier series; summability; conjugate trigonometric series; absolute convergence of Fourier series; series with monotone coefficients; lacunary series; convergence, divergence and absolute convergence of general trigonometric series; problems of uniqueness; representation of a function by a trigonometric series