Fourier transform

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

$$\begin{array}{}\text{(eq:f2lpi)}& f(x)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{e}^{ikx/l}\end{array}$$

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

$$\mathrm{supp}(f)=\{x\in \mathbb{R}\mid f(x)\ne 0\}.$$

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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 $x\le -l\pi$ or $x\ge l\pi$. Then the Fourier coefficients of $f$ (extended with period $2l\pi$) can be given by

$$c_k=\frac{1}{2\pi l}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-ikx/l}dx.$$

Let us define the function $\hat{f}$ by

$$\begin{array}{}\text{(eq:ft)}& \hat{f}(\xi )=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-i\xi x}dx(\xi \in \mathbb{R}).\end{array}$$

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),

$$f(x)=\frac{1}{\sqrt{2\pi }}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}\left(\frac{k}{l}\right)e^{ikx/l}\frac{1}{l}(|x|\le l\pi ).$$

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 $l\to \mathrm{\infty }$ (i.e., $\delta {\xi }_k\to 0$), we expect to have the integral

$$\begin{array}{}\text{(eq:ftinv)}& f(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{i\xi x}d\xi (x\in \mathbb{R}).\end{array}$$

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.