Lacunary trigonometric sum and probability
نویسندگان
چکیده
منابع مشابه
Lacunary Trigonometric Series. Ii
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]...
متن کاملOn 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 ...
متن کاملNo. 7] 5d~ 10. on Lacunary Trigonometric Series
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...
متن کاملPartial Fraction Decompositions and Trigonometric Sum Identities
The partial fraction decomposition method is explored to establish several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation. 1. Outline and introduction Recently, in an attempt to prove, through the Cauchy residue method, Dougall’s theorem (Dougall [6, 1907], se...
متن کاملA Lower Bound in the Tail Law of the Iterated Logarithm for Lacunary Trigonometric Series
The law of the iterated logarithm (LIL) first arose in the work of Khintchine [5] who sought to obtain the exact rate of convergence in Borel’s theorem on normal numbers. This result was generalized by Kolmogorov [6] to sums of independent random variables. Recall that an increasing sequence of positive numbers {nk} is said to satisfy the Hadamard gap condition if there exists a q > 1 such that...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Tohoku Mathematical Journal
سال: 1970
ISSN: 0040-8735
DOI: 10.2748/tmj/1178242716