Book Review: Trigonometric series, Vols. I, II
Jean‐Pierre Kahane · Bulletin of the American Mathematical Society · 2004
J.E. Littlewood called it the Bible.After so many years, it is more the Bible than ever: it is a message of permanent value, an absolute chef-d'oeuvre, and a reference book for now and for years to come.This "third edition", apart from an illuminating foreword by Robert Fefferman as an heir of Zygmund's "Chicago school", is just a reproduction of the second edition of the book, printed first in 1959 and again in 1968 and 1977.The first edition was published in 1935 as volume V of the Polish series Monografje Matematyczne and was entitled Trigonometrical Series.Trigonometrical Series was a much smaller book than Trigonometric Series, but it was already full of content, methods, results, and ideas, all expressed in a pure and rigorous style.Let us compare these two books, the Old Bible and the New One.It gives us an opportunity to enter the history of the subject.Trigonometrical Series was the work of a young man vigorously involved in the renewal of the subject that took place in the years 1900-1930.In 1900, Fejér's theorem gave a new look to the theory of Fourier series; indeed, there were strange phenomena about convergence, namely, the example of du Bois Reymond of a continuous function whose Fourier series diverges at a given point, but the simplest summability method, by arithmetical means, avoided all difficulties of that kind.Summability methods, convolutions, positive kernels, and regularization of functions reestablished Fourier series as one of the central subjects in mathematics.The main step in the renewal of the theory, however, was the new concept of an integral by Lebesgue in 1901.The Riemann integral had been introduced in order to give a precise meaning to Fourier's formulas, namely, the computation of coefficients by means of integrals.The Lebesgue integral soon appeared as a much better tool, so that the use of the term "Fourier series" became reserved for trigonometric series whose coefficients are obtained through Fourier formulas in the sense of Lebesgue.In modern notations, L 1 (T) became the natural frame for Fourier series.However, there is no easy characterization of Fourier coefficients in this context.The simple case is L 2 (T), and the Riesz-Fischer theorem (1907) expresses that the Fourier formulas provide an isomorphism between L 2 (T) and ℓ 2 (Z).Both Fischer and F. Riesz used the fact that L 2 (T) is complete (except that the sentence "L p is complete" needed a new set of definitions and was popularized in this form only in 1930 in the book of Banach, Théorie des opérations linéaires, tom I, of the Monograpfje Matematiczne).The Lebesgue integral established a strong interplay between trigonometric series, integration, derivation, functions of a real variable (this already appeared with Lebesgue), functions of a complex variable (Fatou), 2000