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?



Regularity of sparse Fourier transforms
If $0 < \alpha < 1/2$ then a continuous function on the circle is $\operatorname{Lip}\_\alpha$ only if the Fourier coefficients satisfy $a\_n = {\rm O}( n^{-\alpha})$; this is in Katznelson's book (Chapter I, Corollary 4.6) for instance.


[*EDIT (2013-07-10)*: at the time I thought this was "iff" but a comment points out that I misremembered; in any case, for lacunary series such as the one in the question, a lot more is known than in the general case; see e.g. Katznelson Chapter V for the basics.]


So the function you defined above isn't going to be Hölder continuous for any positive exponent, even though it's clearly continuous (absolutely convergent Fourier series).


Off the top of my head, I don't know of any particularly good source for the higher-dimensional stuff.


