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 ...
متن کاملSome Generalized Lacunary Power Series with Algebraic Coefficients for Mahler’s U−number Arguments
In this work, we show that under certain conditions the values of some generalized lacunary power series with algebraic coefficients for Mahler’s Um−number arguments belong to either a certain algebraic number field or ⋃t i=1 Ui in Mahler’s classification of the complex numbers, where t denotes a positive rational integer dependent on the coefficients of the given series and on the argument. Mo...
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]...
متن کاملOn lacunary series with random gaps
We prove Strassen’s law of the iterated logarithm for sums ∑N k=1 f(nkx), where f is a smooth periodic function on the real line and (nk)k≥1 is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.
متن کاملA note on lacunary series in $mathcal{Q}_K$ spaces
In this paper, under the condition that $K$ is concave, we characterize lacunary series in $Q_{k}$ spaces. We improve a result due to H. Wulan and K. Zhu.
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ژورنال
عنوان ژورنال: Michigan Mathematical Journal
سال: 2006
ISSN: 0026-2285
DOI: 10.1307/mmj/1144437438