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Math Help - lacunary fourier series

  1. #1
    who
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    lacunary fourier series

    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?
    Last edited by who; December 15th 2010 at 05:15 AM.
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  2. #2
    who
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    Doesn't anybody have an idea?
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