# convergence of a series involving cosines

Question:

1) How to determine the convergence of

$\displaystyle \sum_{k=1}^{\infty} \frac{\cos(k^{\alpha} x)}{k^{\alpha}} (-1)^k$

where $x \in \mathbb{R}$ and $\alpha \in (0,1]$. I am especially interested in the case of $\alpha = 1/2$.

2) For a fixed $\alpha$, if the above series converges for every $x$, is the convergence uniform? Is the resulting sum bounded in $x$?

I found the series tests (alternating test,etc.) I learned not useful in this situation, except that the convergence is clear for $x = 0$...

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Looks like a lacunary Fourier series. Have you seen projecteuclid.org/euclid.bams/1183525927 already? –  J. M. Aug 24 '10 at 6:19
...and this: archive.numdam.org/ARCHIVE/CM/CM_1962-1964__15_/… too; Kolmogorov's theorem seems to be the key. –  J. M. Aug 24 '10 at 6:21
thanks, but i thought lacunary means $a_k$ grows exponentially? here $k^{\alpha}$ does not. –  mr.gondolier Aug 24 '10 at 6:29
Have you tried things like the Van-Corput method? It should be applicable here ... –  Helge Aug 24 '10 at 7:11
@ Helge: Can you give me a reference please? I never heard of it before... –  mr.gondolier Aug 24 '10 at 7:29