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Commutative Harmonic Analysis BOLYAI SOCIETY A Panorama of Hungarian Mathematics MATHEMATICAL STUDIES, 14 in the Twentieth Century, pp. 159–192. Commutative Harmonic Analysis JEAN-PIERRE KAHANE The present article is organized around four themes: 1. the theorem of Fej´er, 2. the theorem of Riesz–Fischer, 3. boundary values of analytic functions, 4. Riesz products and lacunary trigonometric series. This does not cover the whole field of the Hungarian contributions to commutative harmonic analysis. A final section includes a few spots on other beautiful matters. Sometimes references are given in the course of the text, for example at the end of the coming paragraph on Fej´er. Other can be found at the end of the article. 1. The theorem of Fejer´ On November 19, 1900 the Acad´emiedes Sciences in Paris noted that it had received a paper from Leopold FEJEV in Budapest with the title “Proof of the theorem that a bounded and integrable function is analytic in the sense of Euler”. On December 10 the Comptes Rendus published the famous note “On bounded and integrable fonctions”, in which Fej´ersums, Fej´er kernel and Fej´ersummation process appear for the first time, and where the famous Fej´ertheorem, which asserts that any decent function is the limit of its Fej´ersums, is proved. The spelling error of November 19 in Fej´er’sname (Fejev instead of Fej´er)is not reproduced on December 10. It is replaced by an other one: the paper presented by Picard is attributed to Leopold TEJER. This is how Fej´er’sname enters into history (C.R. Acad. Sci. Paris, 131 (1900), 825 and 984–987). Fej´erwas a very young man, 20 years old, and unknown. But Fej´er’s theorem became famous quickly. It was used for proving the completeness of the trigonometric system in Hurwitz’s work on the isoperimetric problem, 160 J.-P. Kahane extended by Lebesgue to Lebesgue-integrable fonctions (Fej´er–Lebesgue the- orem), generalized to other kernels useful in approximation theory (Ch. de la Vall´eePoussin), applied in a simple and completely new proof of the Dirichlet–Jordan theorem (Hardy), made more precise by the notion of ab- solute summability (Hardy and Littlewood), and, above all, its simplicity made it accessible to any mathematics student f1g, f2g, f3g, f4, p. 245g, f5g, f6g. By the 1910’s the Fej´ertheorem had already the status of a classi- cal result, and no mathematician could ignore Fej´er’sname and its spelling — perhaps except for the place of the accent. The change can be appreciated in comparing the first and the second edition of Ch. de la Vall´eePoussin’s Cours d’Analyse f7g. In the first edi- tion (1903–1906) it follows the tradition of the great analysis textbooks of the time: it devotes little place to Fourier series and presents Dirichlet’s convergence theorem in the tradition of Dirichlet and Jordan. In the second edition (1912) we can find the Fej´ertheorem, Hardy’s method for deducing the Dirichlet–Jordan theorem from it, and also Fej´er’sexample of a contin- uous functions whose Fourier series diverges at one point. The difference shows well enough the importance of the revolution which took place. What was the nature of this revolution? In order to see this, let us go back to 1900. In the 19th century the theory of analytic functions of one complex variable progressed by giant leaps. It had become “Theory of Functions” par excellence. The theory of functions of several real variables had developed through the investigation of partial differential equations arising in physics. On the other hand, the field of functions of one real variable revealed strange and disquieting creatures: continuous but nowhere differentiable functions (Weierstrass), continuous functions whose Fourier series diverges at one point (du Bois Reymond). Hermite wrote to Stieltjes that he “turned away with fright and horror from that lamentable ulcer: a continuous function with no derivative”. Poincar´ein l’Enseignement Math´ematiquecomplained that examples were not constructed any more in order to illustrate theorems and theories, but just for the purpose of showing that our predecessors were wrong. Gaston Darboux, who wrote an important memoir on discontinuous functions in 1875, turned to geometry quite prudently. Between 1880 et 1900 there were only a few works on trigonometric series and they did not attract much attention. Fourier series did not appear as a reliable tool; there were too many strange things about them. Maybe there are continuous functions whose Fourier series diverge everywhere, just as there are nowhere differentiable continuous functions (this was still an open Commutative Harmonic Analysis 161 question in 1965, before Carleson’s theorem f8g). It is even possible that the Fourier series converges but does not represent the function, as it is the case for Taylor series of C1-functions? This question seems to have been raised by Minkowski ([40, I, p. 24]). Emile Picard’s Trait´ed’analyse, which appeared in 1891, is quite in- structive in this respect. The problem of finding a harmonic function in a domain when boundary values are given (the Dirichlet problem) is treated for the sphere, in the section of the book devoted to functions of several variables, before being treated for the circle. This is not because things are worse for the circle than for the sphere. But the solution for the circle belongs to the chapter “Fourier series” and this is not a subject to begin with f9g. However, the pages devoted by Emile Picard to the Dirichlet problem on the circle are quite interesting; in 1900 Fej´erknew them well and refers to them in his note. Picard presents the method of Schwarz (1872), which is based on the properties of the “Poisson kernel” 1 ¡ r2 X1 = 1 + 2 rn cos nt 1 ¡ 2r cos t + r2 1 As an application he shows that a continuous function on the circle whose Fourier series is X1 (an cos nt + bn sin nt) 0 can be expressed as a uniform limit of trigonometric polynomials of the form X n (an cos nt + bn sin nt)r ; n·N(r) providing therefore a new proof of the Weierstrass approximation theorem. In 1893, Ch. de la Vall´eePoussin also used the Poisson kernel in order to establish the Parseval formula, which is essentially equivalent to the totality of the trigonometric system f10g. In 1901 Adolph Hurwitz stated the Parseval formula as a lemma to his solution of the isoperimetric problem by means of Fourier series, saying that he would prove it later f11g. There were actually a few interesting results on Fourier series, but results and problems were not related to each other. The problem of Minkowski could have been solved easily using the method of Schwarz presented in 162 J.-P. Kahane Picard’s book, but Picard did not know about the problem and Minkowski was apparently unaware of this part of the works of Schwarz and Picard. Schwarz’s method also yielded the formula obtained by Ch. de la Vall´ee Poussin, but de la Vall´eePoussin was not aware of it. Certainly Hurwitz did not know the paper of de la Vall´eePoussin: this is acknowledged in his article of 1903 f1g. Thus all these were isolated works on a marginal subject. Fej´erwrote at the beginning of his thesis that nothing essentially new appeared on Fourier series between 1885 and 1900. Though this is not completely true, still Fourier series appeared as a stagnant subject, out of fashion. Fej´er’sdiscovery is that the averages of the partial sums 1 σ = (S + S + ¢ ¢ ¢ + S ) n n 0 1 n¡1 approximate the given function f at each point where f(x + 0) and f(x ¡ 0) 1 ¡ ¢ exist and f(x) = 2 f(x+0)+f(x¡0) , and uniformly when f is continuous on the circle. Let us trace the circumstances of that discovery. The idea of assigning a sum to a divergent series by means of some summation process was familiar to mathematicians. According to Lebesgue, d’Alembert already used the process of taking averages of partial sums for the series 1 X1 + cos nt (0 < t < 2¼): 2 1 Summation processes became a significant topic of Abel’s investigations, then of those of Poisson, Frobenius, H¨older,Ces`aroand Borel. Abel proved the famous theorem asserting that X1 X1 n lim anr = an r"1 1 1 whenever the right-hand side exists in the usual sense. Poisson considered the left hand side as the generalized sum of the series, whether or not the series converges. Frobenius generalized Abel’s theorem by showing that X1 n 1 lim anr = lim (S0 + S1 + ¢ ¢ ¢ + Sn¡1) r"1 n!1 n 1 Commutative Harmonic Analysis 163 whenever the right side exists. H¨oldergeneralized this theorem of Frobenius by iterating the process of arithmetic means. Ces`arogeneralized another theorem of Abel, on multiplication of series, and introduced for this purpose the processes that we now denote by (C; k). Emile Borel defined new sum- mation processes and applied them to the analytic continuation of functions defined by a Taylor series. Fej´erknew all these works. In order to solve the Dirichlet problem for the circle in the form a X1 f(r cos t; r sin t) = 0 + (a cos nt + b sin nt)rn 2 n n 1 when the function prescribed at the boundary has the Fourier series a X1 0 + (a cos nt + b sin nt) 2 n n 1 one might think of using Abel’s theorem. However, the example of P. du Bois Reymond excludes any hope to obtain a solution for an arbitrary continuous function in this way.
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