The enticement of the Fourier Transform. A Comprehensive Overview, Foundations and Applications

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(Abstract) The Fourier Transform stands as a cornerstone in signal processing, offering profound insights into the frequency content of signals across various domains. This article provides a comprehensive overview of the Fourier Transform, focusing on its theoretical foundations and classical (formal) applications in the computation of integrals, series, and the solution of PDEs and IEs. Beginning with the fundamental concepts, we delve into the mathematical underpinnings of the Fourier Transform, elucidating its ability to bridge, for a given signal, it's time domain $f(t)$ with it's frequency domain $\hat{f}(s)$. Through illustrative examples and explanations, we aim to demystify the Fourier Transform, making it accessible to an extensive audience. Ultimately, this article serves as a valuable resource for students, researchers, and practitioners seeking to grasp the essence and utility of this transformative mathematical tool. Some miscellaneous but interesting applications are also commented at the end of this article (Sections 5.5, 5.6), followed by some special problems that could potentially help the reader to get acquainted with the displayed contents.

(Background knowledge is not provided in this web version)

Instead of announcing the definition of the Fourier transform in a direct manner, we may start with the different features we know so far in order to, somehow, introduce this chapter. As expected, the reference feature is the Fourier series representation. More specifically, the results involving point-wise convergence working with a function of minimal class ($C^1$ works perfectly, but this hypothesis can be weaken).

Let's get to work. Define some function $f:\mathbb{R} \to \mathbb{R}$, not necessarily periodical, and let $f_T : (-T,T] \to \mathbb{R}$ be presented as the signal via restriction $f_T := f|_{[-T,T)}$, for $T>0$. As commented before, if $f$ is a function of first class in the whole real line, we can provide the representation

$$f_T (t) = \sum_{k\in \mathbb{Z}} \left [ \frac{1}{2T} \int_{-T} ^T f(\xi) e^{-\frac{ik\pi \xi}{T}} \mathrm{d}\xi \right ] e^{\frac{ik\pi t}{T}}$$

valid for $-T<t\leq T$. For the notation ahead, we may rewrite this as

$$f_T (t) = \sum_{k\in \mathbb{Z}} \Delta_T \left [ \int_{-T} ^T f(\xi) e^{-2\pi i f_k \xi} \mathrm{d}\xi \right ] e^{2\pi i f_k t}$$

where $\Delta_T := (2T)^{-1}, f_k := k\Delta_T$. The $\Delta$ notation helps us identify the previous expression as a \textit{half} summation for some Riemann integral in $\mathbb{R}^2$ of the fully improper kind. Indeed, taking $T\to +\infty$ the restrained signal converges to the initial data $f$, which must have the expression

$$f(t) = \int_{-\infty}^{+\infty} \left [ \int_{-\infty} ^{+\infty} f(\xi) e^{-2\pi i s\xi} \mathrm{d}\xi \right ] e^{2\pi i st} \mathrm{d}s = \int_{\mathbb{R}^2} f(\xi) e^{2\pi i s (t-\xi)} \mathrm{d}A(\xi ,s) \quad (1)$$

for every $t\in \mathbb{R}$. The integral in brackets is what we are going to define as Fourier transform of $f$ at $x=s$. Also, the second expression for $f(t)$ in (1) is known as the Fourier integral Theorem, which will allow us to consider the Fourier transform as an isometric bijection under some assumptions. We will provide and prove all of these properties more rigorously in this section of the article, albeit not at the very moment.

Definition (Fourier Transform in $L^1$)

Let $f\in L^1 (\mathbb{R})$. As said, we define the (continuous) Fourier transform of $f$,  $\hat{f}: \mathbb{R} \to \mathbb{R}$ as $$\hat{f}(x) = \int_{-\infty} ^{+\infty} f(\xi) e^{-2\pi i x\xi} \mathrm{d}\xi \qquad (2)$$

In multiple cases we will denote the Fourier transform of $f$ as $\mathfrak{F}[f]$, specially when we express $f$ via some functional relation, such as convolutions or compositions, with other functions. We can rapidly make sense of the previous definition, since given any function $f\in L^1 (\mathbb{R})$ one is able to bound (2) directly by

$$\int_{-\infty} ^{+\infty} |f(\xi) e^{-2\pi i x\xi}| \mathrm{d}\xi = \int_{-\infty} ^{+\infty} |f(\xi)|\mathrm{d}\xi = \|f \|_1 < \infty$$

so the integral defining $\hat{f}$ converges absolutely and $\hat{f}$ is well defined in $\mathbb{R}$. Analogously to the case of the Fourier series expansion, in the Fourier transform two different components are pretty well distinguished.

If $f\in L^1(\mathbb{R})$ is an even function, we may notice

$$\begin{eqnarray}\hat{f}(x) & = & \int_{-\infty} ^0 + \int_0 ^{+\infty} f(\xi) e^{-2\pi i x\xi} \mathrm{d}\xi = \int_0 ^{+\infty} f(\xi) \left [e^{2\pi i x\xi} + e^{-2\pi i x\xi} \right ] \mathrm{d}\xi \nonumber \\ & = & \nonumber 2 \int_0 ^{+\infty} f(\xi) \cos (2\pi x\xi) \mathrm{d}\xi =: 2\mathfrak{F}_c [f](x) \end{eqnarray}$$

where the change of variables $-\xi \mapsto \xi'$ has been performed in the section $(-\infty,0]$.

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