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

##### Keywords:
Approximation and series expansion
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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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