lacunary fourier series

**who** · December 15th 2010, 03:33 AM

Hi,
I need some help with the following problem:

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

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?

**who** · December 17th 2010, 06:41 AM

Doesn't anybody have an idea?

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