I have the following formula for the Fourier series of a integrable function $g:[0,N] \to \mathbb{C}$
$$ g(x) = \sum_{n=-\infty}^\infty c_n e^{in2\pi x/N}. $$
I was able to derive the formula for the coefficients $c_n$, but what if a want to show that any integrable function can be actually approximated by the formula given above? Is there something like a theorem?
Best Answer
The statement as stated in the title of the question is false. An arbitrary integrable function $g$ on $[0,N]$ does not necessarily have a Fourier series that converges pointwise to $g$. Precisely how badly this can fail was a major research topic of harmonic analysis well into the later 20th century, and the situation is largely summarized in the links Dr. MV has helpfully provided.
In simpler terms than the Wikipedia article, here is a summary of the situation:
Arbitrary integrable functions on $[0,N]$ are not necessarily pointwise well-approximated by their Fourier series. In fact, the situation is as bad as it could possibly get: Kolmogorov showed that there is a continuous function whose Fourier series diverges on a set of full measure (and in particular, diverges on a dense set).
With an alternative integrability condition, things get better. The Carleson-Hunt theorem states that if $1<p<\infty$ and $g$ is $p$-integrable, i.e. if $|g|^p$ is integrable, then its Fourier series converges to $g$ almost everywhere (i.e. approximates $g$ well, except on a negligible set of points).
If $g$ is piecewise continuous and has a jump discontinuity at $x_0$, then the Fourier series approximation tends to overshoot the actual value of the function from both directions by about $9\%$. This is known as the Gibbs phenomenon, and is of importance in signal processing. In particular, this overshoot will happen at the endpoints $0$ and $N$ unless $g(0) = g(N)$.
Better convergence results can be obtained by prescribing better regularity than merely integrability and continuity. For example, if $g$ is integrable and Holder continuous of order $\alpha\in(frac{1}{2},1]$ (basically slightly better than mere continuity), then the Fourier series of $g$ will converge uniformly to $g$. In particular, if $g$ is continuously differentiable on $[0,N]$, or even merely differentiable with bounded derivative, then the Fourier series of $g$ will converge uniformly to $g$.
If one replaces the usual notion of convergence with an alternative convergence criterion, then one can obtain much better results. For example, it is well-known that if $g$ is square-integrable, then its Fourier series converges to $g$ in mean-square sense. If $g$ is integrable, then its Fourier series is Cesaro summable and Abel summable almost everywhere and in mean to $g$, and the convergence is uniform if $g$ is continuous. However, these notions of convergence do not imply pointwise convergence, and in the case of Cesaro and Abel summability are slightly weaker criteria.