Fourier Series and Topological Groups: Mathematical Decomposition of Periodic Functions

In general, we can assume a period of $2\pi$, i.e., $f(x + 2\pi) = f(x + 2\pi)$. From classical analysis, we know that every periodic function can be decomposed into the sum of its harmonics through its Fourier series, i.e., $$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}$$ (37) where the $c_n$ are complex coefficients that are easily obtainable. This series is a decomposition into harmonics of multiple frequencies: the terms of the summation $\{c_n e^{inx}\}$ are called Fourier modes and the coefficients $c_n$ identify the amplitude of the harmonics. Already at this point, we can notice the hidden importance of the topological nature of the function's domain. Indeed, if the function is periodic with period $2\pi$, equivalently it can also be thought of as a function defined on a unit circle. The coefficients $c_n$ of the Fourier series are in fact obtained simply from a kind of weighted average of the function defined on the circle, i.e., $$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)e^{-inx}dx$$ (38) This decomposition into Fourier modes is the mathematical translation of the musical acoustic phenomenon that we previously described with the vibrating string. If in the previous paragraph we had to insist on the fact that the two ends of the string were fixed, here we had to insist on the periodicity of the function. In both cases, the requirement made was equivalent to requiring that our function be defined on a circumference. Now we will use some theorems of contemporary mathematics to generalize this acoustic physics phenomenon to other phenomena and contexts modeled through a group $G$ that maintains the mathematical structure of the circumference unchanged. What it means to keep the mathematical structure of the circumference unchanged is somewhat ambiguous; we could require that there exists an isomorphism between $G$ and the unit circle of the complex plane equipped with the usual multiplication, i.e., $$S^1 = \{z \in \mathbb{C}: z = e^{i\theta}, \theta \in \mathbb{R}\}$$ (91) However, a more accurate study of the circle $S^1$ allows us to identify two fundamental characteristics, which are those that effectively generalize the concept of circumference to the maximum: the compactness of $S^1$ and its abelianity, i.e., $z_1 z_2 = z_2 z_1$. (39) If previously we had focused on the acoustic phenomenon of the excitation of a string and $G$ indicated a vibrating string with two fixed ends, this time $G$ can equivalently be the rim of a glass, the Earth's orbit around the Sun, a relativistic vibrating string, the day/night temporal cycle, a cellular biological cycle, and more generally any phenomenon that can be modeled through the use of a compact and abelian topological group. Given $G$ a compact and abelian topological group, let us now consider the Pontryagin dual of $G$, namely $\hat{G}$ as the group of homomorphisms from $G$ to the previously defined circle $S^1$, i.e., $\hat{G} = \text{Hom}(G, S^1)$. If $G$ is compact, then it can be shown that $\hat{G}$ is discrete. In the musical acoustic phenomenon considered, the string was excited through a pluck that was then decomposed into harmonics or normal modes or Fourier modes. The mathematical tool to generalize the concept of mechanical excitation of the string in our example will be that of considering square integrable functions on the group, i.e., $L^2(G)$. It can be shown that an arbitrary element $f \in L^2(G)$ can be decomposed as a function of elements of the Pontryagin dual. If the group $G$ is compact, then, since $\hat{G}$ is discrete, the element $f$ will be obtained through the summation of elements $\hat{f}(\gamma)$ which play the role of the Fourier coefficients of the previous example and which are obtainable through $$\hat{f}(\gamma) = \sum_{x \in G} f(x)\gamma(x)$$ (40) where $\gamma \in \hat{G}$. It is therefore important to understand that for any compact and abelian topological group there exists a perfect equivalent of the decomposition into harmonics and Fourier series which is in effect deeply connected to the topological and geometric properties of the circle. If we relax the conditions of compactness and abelianity, we have similar but not equivalent results. If we replace the requirement of an abelian and compact topological space with one that is abelian and only locally compact, then we will have that the described results remain unchanged but the Pontryagin dual will no longer be discrete, but continuous even if compact. This means that we will no longer have a number of harmonics in correspondence with the Natural Numbers, but a continuous number of harmonics to take into account. Conversely, if we give up on the abelianity of the group, it will be possible to have something similar through the Peter-Weyl theorem, where however we will have to consider, instead of the representations of $G$ on the circle $S^1$, all the representations.