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Some properties of trigonometric series whose terms have random signs. (English) Zbl 0056.29001
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Approximation and series expansion
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  Cf., in particular,Paley andZygmund,Proc. Cambridge Phil. Soc., 26 (1930), pp. 337–357 and 458–474, and 28 (1932), pp. 190–205.  We write brieflyR n(x) p.p. (”presque partout”).  Cf.R. Salem, The absolute convergence of trigonometric series,Duke Math. Journal, 8 (1941), p. 333.  SeeTamotsu Tsuchikura,Proc. of the Japan Academy, 27 (1951), pp. 141–145, and the results quoted there, especiallyMaruyama’s result. · Zbl 0042.30003  SeeBulletin des Sciences Mathématiques, 74 (1950).  SeePaley andZygmund,loc. cit.Proc. Cambridge Phil. Soc., 26 (1930), pp. 337–357, or Zygmund,Trigonometrical Series, p. 125.  Since the exponential function is continuous, the uniformity of convergence is (as is very well known) not indispensable here.  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  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.  In what follows {$$\lambda$$} is always positive.  SeePaley andZygmund,loc. cit., andR. Salem,Comptes Rendus, 197 (1933), pp. 113–115 andEssais sur les séries trigonométriques, Paris (Hermann), 1940.  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.  SeeZygmund,, p. 251.  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.  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.  (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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