
lacunary fourier series
Hi,
I need some help with the following problem:
$\displaystyle
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 $\displaystyle S_N$ are essentially equal to the delayed means $\displaystyle \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 $\displaystyle 2^n$ and $\displaystyle N'\le N$: $\displaystyle S_N(g)=\Delta_N'(g)$
I don't see it, this must mean that $\displaystyle 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?

Doesn't anybody have an idea?