On the law of the iterated logarithm for lacunary trigonometric series, II
نویسندگان
چکیده
منابع مشابه
A 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...
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This was first proved for Bernoulli random variables by Khintchine. Salem and Zygmund [SZ2] considered the case when the Xk are replaced by functions ak cosnkx on [−π, π] and gave an upper bound ( ≤ 1) result; this was extended to the full upper and lower bound by Erdös and Gál [EG]. Takahashi [T1] extends the result of Salem and Zygmund: Consider a real measurable function f satisfying f(x + 1...
متن کاملLacunary 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]...
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A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim supN→∞NDN (nkx)(N log logN)−1/2 ≤ Cq a.e. The value of the limsup is a long standing open problem. Recently Fukuyama expli...
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 ...
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ژورنال
عنوان ژورنال: Tohoku Mathematical Journal
سال: 1975
ISSN: 0040-8735
DOI: 10.2748/tmj/1203529250