On the convergence of lacunary trigonometric series
نویسندگان
چکیده
منابع مشابه
On Lacunary Trigonometric Series.
1. Fundamental theorem. In a recent paper f I have proved the theorem that if a lacunary trigonometric series CO (1) X(a* cos nk6 + bk sin nk9) (nk+x/nk > q > 1, 0 ^ 0 ^ 2ir) 4-1 has its partial sums uniformly bounded on a set of 0 of positive measure, then the series (2) ¿(a*2 + bk2) k-l converges. The proof was based on the following lemma (which was not stated explicitly but is contained in ...
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where E c [0, 1] is any given set o f positive measure and {ak} any given sequence of real numbers. This theorem was first proved by R. Salem and A. Zygmund in case of a -0, where {flk} satisfies the so-called Hadamard's gap condition (cf. [4], (5.5), pp. 264-268). In that case they also remarked that under the hypothesis (1.2) the condition (1.3) is necessary for the validity of (1.5) (cf. [4]...
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x (1.2) lim I {t; t e E, SN(t) < xAN} / I E _ (2~r)-1/2 exp( u2/2)du. *' Recently, it is proved that the lacunarity condition (1.1) can be relaxed in some cases (c.f. [1] and [4]). But in [1] it is pointed out that to every constant c>0, there exists a sequence {nk} for which nk+l > nk(1 + ck--1(2) but (1.2) is not true for ak =1 and E_ [0, 11. The purpose of the present note is to prove the fo...
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It is well known from the classical theory of trigonometric series that the series (1) cannot converge except possibly for values of x in a set of zero Lebesgue measure . Our object is to discover how `thin' this set is for various types of sequence {nk } . The first result of this kind is due to P . Turan who proved, in 1941, that the series (1) converges absolutely in a set of positive logari...
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ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1930
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-16-1-90-107