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...

Suppose $F$ has discrete Fourier transform $(a_n)$ where $a_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a_n=1/k$ (or $a_n=1/k^2$ 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:

$$ F(x)=\sum_{k=1}^\infty k^{-2} \exp(ix2^k) $$

for example.

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

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?

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...

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...