I want to know whether there are a finite number of coefficients in a Fourier series of a periodic function (with period $P$), whose magnitude are above a certain threshold. Those coefficients can can be calculated with,
$$
a_n = \frac{2}{P}\int_{0}^{P}{f(x)\cos\left(\frac{2\pi nx}{P}\right)dx},
$$
$$
b_n = \frac{2}{P}\int_{0}^{P}{f(x)\sin\left(\frac{2\pi nx}{P}\right)dx}.
$$
For most functions $a_n$ and $b_n$ will go to zero when $n$ goes to infinity, such as the series of the square and sawtooth wave. Thus those functions would satisfy the requirement.
An example of a function which does not satisfy this would be,
$$
f(x)=\sum_{n=0}^\infty \cos\left(\frac{2\pi nx}{P}\right),
$$
which seems to be a periodic Dirac delta function (correct me if I am wrong). Is there an easy way to identify whether a function wouldn't satisfy the requirement. I suspect that this would happen for functions which have periodic limits where the function goes to $\pm\infty$. But I do not know how to prove this and whether this would include all functions.
A side question, am I correct that only periodic functions that are continuously differentiable will have a finite number of coefficients which are non-zero?
Best Answer
Fourier series' behavior has absorbed the attention of many people over the last 200 years!
First, for pointwise convergence of the Fourier series with coefficients given by the formula you cite: there is the convergence criterion of Fourier-Dirichlet (often not-quite-accurately attributed to Dirichlet alone). An easy version of this sort of thing is: if the (periodic) function is piecewise continuously differentiable and at breaks has left and right derivatives, then at differentiable points the Fourier series converges (pointwise!!!) to the function.
But (by Baire category arguments) there are many continuous functions whose Fourier series fail to converge to them (pointwise) at (for example) all rational numbers.
Also, there is the discrepancy between $L^2$ (also called "mean-square") convergence and pointwise convergence... neither quite implies the other... although see Carleson's theorem: the Fourier series of an $L^2$ function does converge pointwise to the function off a set of Lebesgue measure $0$.
On the other hand, Kolmogorov gave an example of an $L^1$ function (meaning $\int_0^P |f|<\infty$) whose Fourier series diverges everywhere.
Although many, many more things can be said about convergence behavior in classical senses... since you mention the periodic Dirac delta (also known as the Dirac comb): pointwise convergence is not at all the only sort of convergence we can consider. Already $L^2$ convergence (and $L^p$) is somewhat different, but, still, the Fourier coefficients go to $0$. But the periodic Dirac delta's coefficients do not go to $0$. Thus, we could ask _in_what_sense_ that series converges, that is, in what sense the periodic Dirac delta is some-sort-of limit of the finite partial subsums of that Fourier series. Evidently not as a pointwise-valued function...
But it is the limit of the finite subsums in the weak-dual (also called weak-*) topology of distributions, meaning that for every smooth periodic function $f$, $\lim N\sum_{|n|\le N} 1\cdot \widehat{f}(n)\to f(0)$. A stronger statement is also true, that that Fourier series converges (for example) in the index $-1/2$ $L^2$ Sobolev space, which is a Hilbert space defined for general real index $s$ as $H^s=\{\sum_n c_n e^{2\pi inx}:\sum_n |c_n|^2\cdot (1+n^2)^s<\infty\}$. (More properly, it is the completion of the space of finite Fourier series with respect to that norm.) That is, $H^0$ is the usual periodic $L^2$ space, but/and negative-index Sobolev spaces are types of distributions, as opposed to pointwise-valued functions. Their Fourier series can have coefficients that blow up quite a bit, but although not converging pointwise, they do converge in the Sobolev spaces.
Further, in this context, Fourier series with coefficients allowed to grow polynomially can always be differentiated termwise... so long as we interpret the outcome as possibly lying in a Sobolev space or some other type of space of generalized functions.