I really don't think this should be on MO - is there any real mathematical question here? Many other questions have been closed, even though they were genuine and interesting mathematical questions, just for being too easy and/or standard. So why should this question be open?

I suggest you read the following books for a good introduction: Kehe Zhu, Operator Theory on Function Spaces (both Hardy and Bergman spaces); and P. Koosis, Introduction to $H_p$ spaces; P. Duren, Theory of $H^p$ spaces for the Hardy spaces, etc. There is a more recent book (a yellow Springer grad. text) by Hedenmalm, Korenblum, Zhu, Theory of Bergman spaces, which I haven't read but is probably very good.

John Mangual: it seems you are extremely confused about the basic definitions of the Hardy space, Bergman space, and Bergman kernel. The Bergman space, not the Hardy space, is what is relevant for the Bergman kernel; the $\rho$ you've written down is not the Bergman kernel; and the Bergman kernel is a function, not a projection operator. An orthogonal basis for the Hardy space $H^2$ or the Bergman space $L^2_a$ is very easy: just take $\{1, z, z^2, \ldots\}$ and normalise if you wish.

Why not just define the function to be zero outside the interval? Or, if the interval is bounded, why not use Fourier series instead? You need to specify in more detail why you want to use the Fourier transform; it changes certain functions into certain other functions, but without knowing what properties you seek, it is difficult to know exactly what you should be looking for.

silmaril89: are you sure you don't mean $y-x$ instead of $(y-x)^2$ in the integral? That would make a bit more sense, because then the positive/negative parts of $y-x$ cause some cancellation; look up the Hilbert Transform. See also my comment to Michael Renardy's answer below.

– silmaril89: The $1/(y-x)^2$ term is always positive and diverges when integrated through $x$, so if $\phi$ is continuous then $\phi(x)$ must be zero for every $0<x<1$ for the integral to be finite. If you don't assume $\phi$ is continuous, you get into murkier territory involving measure theory, Lebesgue points etc. but I expect you'd still get $\phi = 0$ almost everywhere. This is why I think you've mistyped the equation; see my comment above.

...but of course, these are only very small values of c, so it would be very dangerous to take such limited evidence seriously!

I searched for best approximations up to about c<420; the values 32, 69, 98, 132, 181, 210, 301, 373 for c are better than nearby values. A linear regression of log(error) on log(c) gave the best fit line log(error) = -4.23417 log(c) -5.7329, with correlation coefficient -0.986. Doubtless, someone more skilled with computers with fancy graphical software could do a much better analysis; this seems to suggest an exponent of around 4 rather than 6.

Sorry to be so stupid; where does your $1/q^6$ come from? From $(1/q^2)^3$, or something deeper?

Former world champion Dr. Botvinnik was also an electrical engineer.

Well, in my college chess teams, most of the chess players were studying mathematics or computer science; but it was very noticeable that the best mathematicians were only average chess players, and vice versa, with few exceptions. (And, several of the top mathematicians didn't play at all). Former World Chess Champion Karpov gave up mathematics at university because mathematics and chess were "incompatible". These are not statistically significant samples, of course!

@Andres: OK, I see. That's interesting, but different to your question as I understood it; your question says "What is known about those sets...", without explicitly saying that you are assuming the sets to have measure zero. So it seems (to me) like you want a refinement of the Kahane/Katznelson result, and the Carleson result is a separate side-issue.

@Andres: sorry to keep leaving all these comments, but you say: "I imagine the question here can be answered by direct constructions that do not require the use or knowledge of the theorem". But, suppose you had the desired THEOREM: $E$ is a set of divergence for some $f$ if and only if...(DIV) The hypothetical criteria (DIV) must somehow imply zero measure (maybe non-trivially). However, for (DIV) to be a useful, usable criterion, we would expect it to be possible to prove directly that (DIV) for $E$ implies $m(E) = 0$. This would then give the Carleson theorem, wouldn't it?

...but of course, the Fourier series question here is genuinely more general than for power series, I think. You can split any Fourier series into two power series in $z$ and $\bar{z}$ and apply the results for power series, but it's possible (maybe) for each to diverge, but the sum to converge.

@Andres: I've just noticed the other question "Behaviour of power series on their circle of convergence" you referred to above [together with your wonderful answer!] That question, of course, is just the special case of this question where the Fourier coefficients for negative $n$ are all zero. It seems like you've almost answered your own question!

...Of course, for continuous $f$ at least.