# Math Help - lacunary fourier series

1. ## lacunary fourier series

Hi,
I need some help with the following problem:

$
g(x):=\(\sum \limits_{k=0}^{\infty} 2^{-k/2}exp(i2^k x)
$

This is a lacunary fourier series which means that the series skips many terms. Because of this lacunary property the partial sums $S_N$ are essentially equal to the delayed means $\Delta_N' (g)=S_{N'} + 2\(\sum \limits_{k=N'}^{2N'} (1- \frac{|k|}{2N'})2^{-k/2}exp(i2^k x)$
If N' is the largest integer of the form $2^n$ and $N'\le N$: $S_N(g)=\Delta_N'(g)$

I don't see it, this must mean that $2\(\sum \limits_{k=N'}^{2N'} (1- \frac{|k|}{2N'})2^{-k/2}exp(i2^k x)=0$, doesn't it? Why would this be true? Can anyone help me please?

2. Doesn't anybody have an idea?