Series help, fourier series How do I know if a given function can be represented by a fourier series, that converges to the value of that function at non discontinuities. Also where did Fourier come up with the idea of representing periodic function in the way he did?
 A: There's various sufficient conditions, depending on how technical you want them to be. You obviously want $f$ to be periodic no matter what, I'll assume that from now on. Dirichlet's Theorem on convergence of Fourier series states that if both $f$ and its derivative are piecewise continuous on $[0,2\pi]$ (or whatever interval its periodic on, you can always rescale and translate a periodic function to get one periodic on $[0,2\pi]$), then the Fourier series of $f$ always converges to $f(x)$ when $f$ is continuous at $f$, and to the average of the left and right limits at $x$ of $f$ when $f$ is discontinuous.
Carleson's Theorem states that if $f$ is square-integrable over $[0,2\pi]$, that is
$$\left(\int_{0}^{2\pi}{|f(x)|^{2}\ dx}\right)^{\frac{1}{2}}<\infty,$$
then the Fourier series of $f$ at $x$ converges to $f(x)$ for almost all $x\in[0,2\pi]$.
A: This is an eminently natural question! In addition to Peter's accurate remarks:
From the (admittedly mostly secondary) historical sources I've read, Fourier did not initially have the inner-product formula for the coefficients, and had no really mathematical argument in favor of the expressibility of periodic functions (and, in those days, what was a "function"?) except analogies from mechanics and "overtones" in vibrating systems. That heuristic, however, set him on a course that seemed very productive for (apparently) solving a certain incarnation of the heat equation. Thus, the (apparent) utility of the idea gave motivation to subsequent legitimization.
In those days, there was no clear sense of "convergence", except (not uniform!) pointwise. Certainly not any formal "$L^2$" convergence. Indeed, problems about pointwise convergence of Fourier series led Cantor to create set theory.
Even with improved vocabulary of modern times, there is considerable tension between the "natural" pointwise convergence and, for example, $L^2$ convergence, or convergence in Levi-Sobolev spaces, or distributional convergence. Arguably, the $L^2$ theory works most smoothly, and, arguably, the $L^2$ theory of Levi-Sobolev spaces gives a more coherent and robust approach to (uniform!) pointwise convergence, if that is truly needed. E.g., while, perhaps counter-intuitively, the Fourier series of a $C^1$ function provably converges to it (uniformly) pointwise, it does not typically converge to it in the $C^1$ norm. (Also, the Fejer kernel discussion, while proving that finite Fourier series are dense in $C^o$, does not at all promise that it is the finite truncations of the Fourier series that converge to the function.) Meanwhile, functions in the ${1\over 2}+\epsilon$ Levi-Sobolev space have Fourier series that converge to them in that topology, and (Levi-Sobolev imbedding thm) are continuous, and the Fourier series also converges in the $C^o$ topology, and so on.
That is, fixation on pointwise convergence as fundamental may be misguided, although we are "brought up" to think of functions as primarily giving pointwise values. :)
A: A classical theorem on pointwise convergence of Fourier series says that if $f(x)$ is piecewise smooth on $(-\ell,\ell)$, then the Fourier series of $f$ converges pointwise on $(-\ell,\ell)$. Moreover, the value to which the Fourier series converges at $x=x_0$ is $${f(x_0^+)+f(x_0^-)\over 2},$$ where the superscripts denote the one-sided limits $$f(x_0^+):=\lim_{x\to x_0^+}f(x)\quad\text{ and }\quad f(x_0^-):=\lim_{x\to x_0^-}f(x).$$ 
In other words, if $x=x_0$ is a point of continuity of $f$, then its Fourier series converges to $f(x_0)$ there, but if $x=x_0$ is a point of (suitable type of) discontinuity of $f$, then its Fourier series converges to the average of the left- and right-and limits of $f$ at $x=x_0$.
