FOURIER SERIES, which are also called trigonometric series, take their name from Jean Baptiste Joseph Fourier (see above), author of La theorie analytique de la chaleur (1822). A great many problems of analytical physics, notably those which relate to the radiation of heat, are easily solved when the initial state is represented by a simple periodic function, as the sine or cosine of the variable or of an integral multiple of the variable. When the initial conditions are entirely arbitrary, it is natural to attempt to represent it by the sum of a finite or infinite number of such simple periodic functions ; i.e., by a trigonometric series. The problem had already been stated by Euler, but before Fourier's time mathematicians had not thought such a repre sentation possible, except for those functions called by Euler continuous functions, i.e., those which may be represented geo metrically by a unique continuous curve. The great discovery of Fourier was the representation of any discontinuous function by a trigonometric series, i.e., by a series of continuous functions. The simplest type of such a discontinuous function is one which is equal to zero in certain intervals and to one in certain other intervals. In this way problems in mathematical physics can be solved for any initial conditions when one knows how to solve for the simple periodic initial conditions.
In his Theorie analytique de la chaleur Fourier indicates also, with respect to the radiation of heat in a sphere, how an arbitrary function might be developed in a series in which the terms are proportional, not to the sines and cosines of the integral multiples of the variable, but to the sines and cosines of the products of the variable and the roots of a certain transcendental equation. One sees in Fourier's work the origin of the theory of orthogonal functions which were further developed at the beginning of the 2oth century by Fredholm in his well-known work on the theory of integral equations. Fourier's discoveries dominate the whole of mathematical physics; the applications and generalizations derived from his methods are far reaching.
The theory of the Fourier series has proved to be no less im portant in mathematical analysis than in physics. The history of its development during the past century is closely knit with the development of the theory of functions of a real variable. Fourier did not give a rigorous demonstration of his results ; as was the custom of the time, they were based upon expansions of a purely formal value. In 1 83 7, however, Lejeune Dirichlet gave a demon stration of Fourier's results that was thoroughly rigorous. Dirich let's essential accomplishment consisted in introducing the idea of uniform convergence of a series, an idea of capital importance in analysis. Following Lejeune Dirichlet, the greatest geometricians of the 19th century occupied themselves with perfecting the theory of the trigonometric series, i.e., of defining exactly the necessary and sufficient conditions that an arbitrary function must satisfy if it can be developed in a series of this nature, the coeffi cients being furnished by the very same formulae stated by Fourier. One important development is due to Riemann, who demonstrated that the convergence of Fourier's series for a given value of a variable depends only on the behaviour of the function in the immediate vicinity of that value. Riemann's theorem re duces the general problem of Fourier's series to the study of the behaviour of a function in the immediate vicinity of a point.
The latest development of the theory of trigonometric series can only be realized by a deeper study of the idea of an aggre gate (ensemble) of points (see AGGREGATES, THEORY OF; POINT SETS). In the same way that Lejeune Dirichlet was inspired by the study of Fourier's series, to perfect his own theory of the con tinuity and uniform convergence of certain series, Georg Cantor, to complete this study, created and perfected the theory of point sets. He describes the aggregates, or sets of points, derived from different orders of a given aggregate or set, and explains that the numbering of these derived sets creates a new type of numbers, to which he gives the name transfinite numbers. Finally, he shows that, if one of these derived sets from the ensemble of discontinu ities of the f unction comprises a finite number of terms, the trigo nometric development is certainly unique. The works of Georg Cantor on the theory of aggregates had an important indirect repercussion on the theory of Fourier's series. They also suggested an exceedingly important question, that of the measure of a linear set of points. This problem of measurement was solved, precisely and definitely, by Emile Borel in 1897, when he described certain aggregates as "measurable aggregates" (ensembles mesurables) and which Henri Lebesgue later called ensembles mesurables B. This definition of measurable aggregates, or sets of points, has, as a consequence, the idea of aggregates, or sets of points, of zero measure. M. Henri Lebesgue has proposed to call a property "almost everywhere true" (veri Presque partout) when it is true of all the points excluding those points of an aggregate of zero measure. This of an "almost everywhere true" property has been important in the recent development of certain trigonometric series.
It is chiefly Lebesgue's idea of the integral, an idea which di rectly attaches itself to the measurement of aggregates, that has originated the latest refinements in the theory of Fourier's series. The coefficients of the series are in fact determined by the definite integrals in which the differential element is equal to the product of the function which is to be represented and the sine or cosine. By replacing integrals according to Riemann's definition by the integrals as defined by Lebesgue, the field of continuous or dis continuous functions which can be described by trigonometric series is considerably extended. These works on the trigonometric series which have their origin in the integral of Lebesgue are too technical and too numerous to be discussed here. A very im portant work by Arnaud Denjoy, however, deserves attention. Thanks to a new generalization of the idea of the integral, which he has named "totalization," Denjoy has been able to solve the problem of the determination of coefficients of a trigonometric series in a complete and definitive manner. At the moment that the series exists and represents a function, the coefficients can be obtained in every case by Fourier's formulae, on condition that the integrals figuring in these formulae are interpreted in the sense given them by Denjoy. This beautiful result definitely closes a series of researches which has extended over more than a century. More recent investigations on a summation of Fourier's series, in a case where the series are divergent, are based for the most part on the methods and the generalizations of Cesaro. For a more detailed study of these researches, see the memoir by Plancherel and the book by Emile Borel, Lefons sur les series divergentes. BIBLIOGRAPHY.-J. B. J. Fourier, Theorie analytique de la chaleur (182 2) and Oeuvres de Fourier (1888) ; Arnold Sachse, "Essais Historiques sur la representation d'une fonction arbitraire d'une seule variable par une serie trigonometrique," Fr. trans. in the Bulletin des Sciences Mathematiques (188o) (Inaugural Dissertation, Gottingen, 1879) ; Henri Lebesgue, Lecons sur les series trigonometriques (19o3) ; E. W. Hobson, Theory of Functions of a Real Variable and the Theory of Fourier Series (1923) ; Plancherel, "Developpement de la Theorie des series trigonometriques dans le dernier quart de siecle," L'Enseigne ment Mathematique (1924) ; Emile Borel, Lecons sur la theorie des fonctions (192 7) and Lecons sur les series divergentes (1928) .
(E. Bo.)