Celebratio Mathematica

R. Salem and A. Zygmund: “Lacunary power series and Peano curves,” Duke Math. J. 12 : 4 (1945), pp. 569–578. MR 0015154 Zbl 0060.20402 article

R. Salem and A. Zygmund: “Capacity of sets and Fourier series,” Trans. Am. Math. Soc. 59 : 1 (1946), pp. 23–41. MR 0015537 Zbl 0060.18511 article

In [1939] A. Beurling proved that if the series \[ \sum n(a_n^2 + b_n^2) \] converges then the points of divergence of the Fourier series \[ a_0/2 + \sum_1^{\infty}a_n\cos nx + b_n\sin nx \] form a set of logarithmic capacity zero. In the same paper, he stated the following result, the proof of which was due to appear, but to our knowledge has not appeared, in the Arkiv für Matematik: if \( 0 < \alpha \leq 1 \) and if the series \[ \sum n^{\alpha-\epsilon}(a_n^2 + b_n^2) \] converges for every \( \epsilon > 0 \), the above Fourier series is summable by Abel’s method except in a set whose \( \beta \)-capacity is zero for every \( \beta > 1 - \alpha \).

The present paper has its origin in the desire of the authors to find a more direct and simpler proof of Beurling’s results and to extend them. In the course of the work, however, some other theorems of independent interest have been established, and are presented here.

R. Salem and A. Zygmund: “The approximation by partial sums of Fourier series,” Trans. Am. Math. Soc. 59 : 1 (1946), pp. 14–22. MR 0015538 Zbl 0060.18510 article

R. Salem and A. Zygmund: “On a theorem of Banach,” Proc. Natl. Acad. Sci. U. S. A. 33 : 10 (October 1947), pp. 293–295. MR 0021611 Zbl 0029.20801 article

R. Salem and A. Zygmund: “On lacunary trigonometric series,” Proc. Natl. Acad. Sci. U. S. A. 33 : 11 (November 1947), pp. 333–338. MR 0022263 Zbl 0029.11902 article

Let us consider a lacunary trigonometric series \begin{equation*}\tag{1} \sum_{k=1}^{\infty}(a_k\cos n_kx + b_k\sin n_kx), \end{equation*} with \( n_{k+1}/n_k > q > 1 \), and let us confine our attention to the interval \( 0 \leq x \leq 2\pi \). Let \( S_N(X) \) denote the \( N \)-th partial sum of (1), that is to say the sum of the terms with \( k = 1, 2, \dots \), \( N \). Let \[ C_N = \bigl\{\tfrac{1}{2}(a_1^2 + b_1^2 + \cdots + a_N^2 + b_N^2)\bigr\}^{1/2} .\] In a recent note [1947], J. Ferrand and R. Fortet considered the behavior of the ratio \( S_N(x)/C_N \). They stated, without proof, that if \( C_N \to +\infty \) as \( N \to\infty \), the distribution function of \( S_N(x)/C_N \) tends to the Gaussian distribution with mean value 0 and dispersion 1. Clearly, the result as stated cannot be correct since, if \[ c_k = (a_k^2 + b_k^2)^{1/2} \] increases very rapidly, the distribution function tends to that of cosine. Thus some kind of restriction on the \( c_k \) is necessary. (Mr. Erdős informs us that a proof of the result, under the condition \( C_n\to\infty \), \( c_n = O(1) \), will be published in a paper written jointly by him, Mlle. Ferrand, Fortet and Kac.) In this note we propose to give a complete solution of the problem.

R. Salem and A. Zygmund: “On lacunary trigonometric series, II,” Proc. Natl. Acad. Sci. U. S. A. 34 : 2 (February 1948), pp. 54–62. MR 002936 Zbl 0029.35601 article

M. Kac, R. Salem, and A. Zygmund: “A gap theorem,” Trans. Am. Math. Soc. 63 : 2 (1948), pp. 235–243. MR 0023937 Zbl 0032.27402 article

Raphaël Salem	Related
Marek Kac	Related

R. Salem and A. Zygmund: “A convexity theorem,” Proc. Natl. Acad. Sci. U. S. A. 34 : 9 (September 1948), pp. 443–447. MR 0027083 Zbl 0034.33501 article

R. Salem and A. Zygmund: “La loi du logarithme itéré pour les séries trigonométriques lacunaires” [The law of the iterated logarithm for lacunary trigonometric series], Bull. Sci. Math. (2) 74 (1950), pp. 209–224. MR 0039828 Zbl 0039.07001 article

R. Salem and A. Zygmund: “Sur les séries trigonométriques dont les coefficients ont des signes aléatoires” [On trigonometric series whose coefficients have random signs], C. R. Acad. Sci. Paris 236 (1953), pp. 571–573. MR 0055478 Zbl 0050.07303 article

R. Salem and A. Zygmund: “Some properties of trigonometric series whose terms have random signs,” Acta Math. 91 : 1 (1954), pp. 245–301. Dedicated to Professor Hugo Steinhaus for his 65th birthday. MR 0065679 Zbl 0056.29001 article

Raphaël Salem	Related
Władysław Hugo Dionizy Steinhaus	Related

R. Salem and A. Zygmund: “Sur un théorème de Piatetçki–Shapiro” [On a theorem of Piatetçki–Shapiro], C. R. Acad. Sci. Paris 240 : part 2 (1955), pp. 2040–2042. MR 0071580 Zbl 0065.29702 article

R. Salem and A. Zygmund: “Sur les ensembles parfaits dissymétriques à rapport constant” [On the asymmetric perfect sets of constant ratio], C. R. Acad. Sci. Paris 240 : part 2 (1955), pp. 2281–2283. MR 0071581 Zbl 0066.31701 article

R. Salem and A. Zygmund: “A note on random trigonometric polynomials,” pp. 243–246 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 2. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1956. MR 0084622 Zbl 0071.28401 incollection

Raphaël Salem	Related
Jerzy Neyman	Related

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