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?