Fourier transform

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So far we have been studying the Fourier series of functions with period \(2\pi\). What about the functions with periods other than \(2\pi\)? Furthermore, what about non-periodic functions? In this section, we consider these problems informally and briefly. 



Suppose \(f\) is a smooth function with a period of \(2l\pi\) where \(l > 0\) is a constant. We have

\[f(x) = \sum_{k = -\infty}^{\infty}c_ke^{ikx/l}\tag{eq:f2lpi}\]

and the right-hand side converges uniformly. The Fourier coefficients are given by

\[c_k = \frac{1}{2\pi l}\int_{-l\pi}^{l\pi}f(x)e^{-ikx/l}dx ~~~ (k \in \mathbb{Z}).\]

Thus, the Fourier series can be readily extended to any periodic function with period other than \(2\pi\).

Next, we consider functions that are not necessarily periodic. Suppose that \(f\) is a function of class \(C^1\) with a bounded support. 

Remark. The support \(\mathrm{supp}(f)\) of a function \(f\) is the subset of the domain, defined by

\[\mathrm{supp}(f) = \{x \in \mathbb{R} \mid f(x) \neq 0\}.\]

□

If the support is bounded, we have

\[\mathrm{supp}(f) \subset (-l\pi, l\pi)\]

for a sufficiently large \(l > 0\). This means that \(f(x) = 0\) if \(x \leq -l\pi\) or \(x \geq l\pi\). Then the Fourier coefficients of \(f\) (extended with period \(2l\pi\)) can be given by

\[c_k = \frac{1}{2\pi l}\int_{-\infty}^{\infty}f(x)e^{-ikx/l}dx.\]

Let us define the function \(\hat{f}\) by

\[\hat{f}(\xi) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-i\xi x}dx ~~~ (\xi \in \mathbb{R}).\tag{eq:ft}\]

Then, we have

\[c_k = \frac{1}{\sqrt{2\pi}}\hat{f}\left(\frac{k}{l}\right)\frac{1}{l}.\]

Substituting this into (eq:f2lpi),

\[f(x) = \frac{1}{\sqrt{2\pi}}\sum_{k=-\infty}^{\infty}\hat{f}\left(\frac{k}{l}\right)e^{ikx/l}\frac{1}{l} ~~~ (|x| \leq l\pi).\]

Note that this is the Riemann sum of the function \(\hat{f}(\xi_k)e^{i\xi x_k}\) if we set \(\xi_k = k/l\) and \(\delta \xi_k = \xi_{k+1}-\xi_k = 1/l\) for \(k = 0, \pm 1, \pm 2,\cdots\). Fixing \(x\) and letting \(l \to \infty\) (i.e., \(\delta \xi_k \to 0\)), we expect to have the integral

\[f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\xi)e^{i\xi x}d\xi ~~~ (x\in\mathbb{R}).\tag{eq:ftinv}\]

What this equation means is that the function \(f(x)\) can be represented as a linear combination of infinitely many \(e^{i\xi x}\) with ``coefficients'' \(\hat{f}(\xi)\).

Given the function \(f\),  \(\hat{f}(\xi)\) defined by (eq:ft) is called the Fourier transform of \(f\). (eq:ftinv) is called the inversion formula of the Fourier transform.

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