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This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!

Suppose $f(x)=\sum_{k\in\mathbb{Z}}c_ke^{ikx}$, whereby we mean pointwise convergence. What properties must $f(x)$ then satisfy? Clearly continuity is too strong (take for example an appropriately defined square wave). $L^1[-\pi,\pi]$ seems troublesome as well, since term-by-term integration is not necessarily valid with only pointwise convergence.

Thanks ahead for any tips!

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@Yemon, what's wrong about the ambient class being all functions $S^1\to\mathbf{C}$? Therein the poster wants to characterize the set of all pointwise limits of pointwise convergent Fourier series. – Francois Ziegler Nov 12 '12 at 0:50
Francois, how is one proposing to define the Fourier coefficients of $f$? Or is the question: "here is a doubly infinite series $(c_n)$ of complex numbers, such that for each $x\in [-\pi,\pi]$ the partial sums $\sum_{|k|\lq N} c_k e^{ikx}$ converge; what can we say about the resulting function $f:[-\pi,\pi] \to {\mathbb C}$?" – Yemon Choi Nov 12 '12 at 1:19
As stated it seems that the question is not related to Fourier series or $L^1$, or any other a priori given class of functions. He seems to ask: suppose that SOME series of the form $\sum c_k\exp(ikx)$ is convergent at every point. What can one say about the sum? Do I understand correctly? – Alexandre Eremenko Nov 12 '12 at 1:37
The context of the question was that of Fourier series, but I think the restatement by @Alexandre is appropriate. – icurays1 Nov 12 '12 at 2:38
It may bear emphasis that various possible senses of the question have wildly different, and not strongly related, answers, and that some bit of confusion about this is evident in the question's language and form. E.g., the Fourier series of a square wave, or other piecewise smooth, but not smooth, function converges not-so-well at the discontinuity. Meanwhile, yes, of course $L^1$ has problems, its problems are arguably less severe than "mere pointwise-valued functions". (Francois Ziegler's cited survey paper is very interesting in the regard of to-me-pathologies, nevertheless!) Clarify? – paul garrett Nov 12 '12 at 3:38
up vote 6 down vote accepted

The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting survey paper on the subject:

One of the problems in the theory of trigonometric series $$\frac12a_0+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\tag{1.1}$$ is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function $f(x)$, then $f(x)$ is necessarily integrable and the coefficients, $a_n$ and $b_n$, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (...) The problem has been solved by Denjoy [4; 5], Verblunsky [14], Marcinkiewicz and Zygmund [10], Burkill [1; 2], and James [8]. (...) The solutions are described, mainly in the order in which they were published, in §§2-7 below.

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Excellent reference, thank you. It appears that the answer is not entirely straightforward, which is good (I thought about this for a while before asking). – icurays1 Nov 12 '12 at 2:37

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