Fourier Transform

$\hat{f}(t):={\int }_{-\infty }^{\infty }f(x)e^{-2\pi itx}dx$, where $f\in L^1(\mathbb{R})$ is an integrable function with respect to the Lebesgue measure.

Intuitive overview

Inverting the Fourier Transform

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Given the Fourier Transform $\hat{f}$ of a function $f$, we are able to retrieve the original function by applying the result $f(x):={\int }_{-\infty }^{\infty }\hat{f}(t)e^{2\pi itx}dt$

Properties

• Fourier Uncertainty Principle

The results below make Fourier transforms very useful

	$f(x)$	$\bar{f}(t)$
Definition	$f(x)$	${\int }_{-\infty }^{\infty }f(x)e^{-2\pi itx}dx$
Inverse	${\int }_{-\infty }^{\infty }\hat{f}(t)e^{2\pi itx}dt$	$\bar{f}(t)$
Linearity	$af(x)+g(x)$	$a\bar{f}(t)+\bar{g}(t)$
Shift	$g(x-x_0)$	$\bar{g}(t)e^{-2\pi it{x}_0}$
Convolution	${\int }_{-\infty }^{\infty }f(x)g(x-y)dy$	$\bar{f}\bar{g}$
Product	$f(x)g(x)$	${\int }_{\infty }^{\infty }\bar{f}(x)\bar{g}(t-x)dx$
Scaling	$f(ax)$	$\Vert a{\Vert }^{-1}f(\frac{t}{a})$
Differentiation	$\frac{d}{dx}f(x)$	$2\pi it\bar{f}(t)$
Integration	${\int }_{-\infty }^{x}f(z)dz$	$\frac{\bar{f}(t)}{2\pi it}$

Look up table

Below we list frequently needed Fourier transforms

	$f(x)$	$\hat{f}(t)$
Dirac	$\delta (x)$	1
Constant	1	$\delta (t)$
Cosine	$\cos (2\pi Ax)$	$\frac{1}{2}(\delta (t-A)+\delta (t+A))$
Sine	$\sin (2\pi Ax)$	$\frac{1}{2i}(\delta (t-A)-\delta (t+A))$
Step function	$1_{x\ge 0}$	$\frac{1}{2\pi it}$