Suppose F has discrete Fourier transform ![(a\sb n)](http://latex.mathoverflow.net/png?%28a%5Fn%29) where ![a\sb n=0](http://latex.mathoverflow.net/png?a%5Fn%3D0) unless ![n=2^k](http://latex.mathoverflow.net/png?n%3D2%5Ek) for some ![k>0](http://latex.mathoverflow.net/png?k%3E0) , in which case ![a\sb n=1/k](http://latex.mathoverflow.net/png?a%5Fn%3D1%2Fk) (or ![a\sb n=1/k^2](http://latex.mathoverflow.net/png?a%5Fn%3D1%2Fk%5E2) if you want: I'm happy with anything polynomial).  What sort of regularity conditions does F have?  Is it Holder continuous, or not?

To be explicit: ![\[F(x)=\sum\sb {k=1}^\infty k^{-2} \exp(ix2^k)\]](http://latex.mathoverflow.net/png?%5C%5BF%28x%29%3D%5Csum%5F%7Bk%3D1%7D%5E%5Cinfty%20k%5E%7B%2D2%7D%20%5Cexp%28ix2%5Ek%29%5C%5D) for example.

More generally, I'm interested in two dimensional (discrete) Fourier transforms: is there a good reference for this sort of thing?

Edit: Slight bug in the sidebar LaTeX converter: it didn't like k>0 as the > gets interpretted as a tag.  Manually changing it to > works...