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Some properties of trigonometric series whose terms have random signs. (English) Zbl 0056.29001

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[1] Cf., in particular,Paley andZygmund,Proc. Cambridge Phil. Soc., 26 (1930), pp. 337–357 and 458–474, and 28 (1932), pp. 190–205.
[2] We write brieflyR n(x) p.p. (”presque partout”).
[3] Cf.R. Salem, The absolute convergence of trigonometric series,Duke Math. Journal, 8 (1941), p. 333.
[4] SeeTamotsu Tsuchikura,Proc. of the Japan Academy, 27 (1951), pp. 141–145, and the results quoted there, especiallyMaruyama’s result. · Zbl 0042.30003
[5] SeeBulletin des Sciences Mathématiques, 74 (1950).
[6] SeePaley andZygmund,loc. cit.Proc. Cambridge Phil. Soc., 26 (1930), pp. 337–357, or Zygmund,Trigonometrical Series, p. 125.
[7] Since the exponential function is continuous, the uniformity of convergence is (as is very well known) not indispensable here.
[8] See also the authors’ notes ”On lacunary trigonometric series” part I,Proc. Nat. Acad., 33 (1947), pp. 333–338, esp. p. 337, and part II,Ibid. Proc. Nat. Acad., 34 (1948), pp. 54–62. · Zbl 0029.11902
[9] Later on we shall need the lemma in the case whenn=and {\(\Sigma\)}c m 2 < It is clear that the inequalities of the lemma hold in this case, since {\(\Sigma\)}c m c m converges almost everywhere.
[10] In what follows {\(\lambda\)} is always positive.
[11] SeePaley andZygmund,loc. cit., andR. Salem,Comptes Rendus, 197 (1933), pp. 113–115 andEssais sur les séries trigonométriques, Paris (Hermann), 1940.
[12] Attention of the reader is called to the fact thatR n, and later onT n, has not the same meaning here as in the preceding chapter.
[13] SeeZygmund,, p. 251.
[14] In particular, the series, {\(\Sigma\)}m (logm)1- m (t) cosmx, for whichR n logn is bounded but does not tend to zero, is not randomly continuous.
[15] We are grateful to Dr.Erdös for calling our attention toChung’s paper. It may be added that (6.1.6.) generalizes an earlier result ofErdös who showed that in the case {\(\alpha\)}1={\(\alpha\)}2=...=1 the left side of (6.1.6) is almost everywhere contained between two positive absolute constants.
[16] (Added in proof.) Dr.Erdös has communicated us that in the casea 1=a 2=...=1 he can prove that, for every {\(\epsilon\)}>0. \(\lim \inf \frac{{M_n \left( t \right)}}{{_n \tfrac{1}{2} - \varepsilon }} > 0\) almost everywhere, and even a somewhat stronger result.
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