On lacunary power series

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 ...

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]...

On 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.

A 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.

Lacunary Fourier Series on Noncommutative Groups