Fourier transforms

1 Incoming

I especially need to learn about Fourier transforms of radial functions.

• The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever? is a virtuosic, illustrated, surprisingly deep explanation of some crucial ideas (specifically the fast Fourier transform) by reducible. Also, great animations.

• Terry Tao is kind of an expert on this and writes nice crisp lecture notes on the idea:

  • 246B, Notes 2: Some connections with the Fourier transform
  • 246B – complex analysis
  • course in Modern real-variable harmonic analysis.

2 Useful properties

$f(x)$

$\hat{f}(\nu )=\int f(x)e^{-i\nu x}dx$

Linearity: $a\cdot f(x)+b\cdot g(x)$

$a\cdot \hat{f}(\nu )+b\cdot \hat{g}(\nu )$

Time shift $f(x-a)$

$e^{-ia\nu }\hat{f}(\nu )$

Frequency shift $f(x)e^{iax}$

$\hat{f}(\nu -a)$

Dilation $f(ax)$

$\frac{1}{|a|}\hat{f}\left(\frac{\nu }{a}\right)$

Duality $\hat{f}(x)$

$2\pi f(-\nu )$

$\frac{d^{n}f(x)}{d{x}^n}$

$(i\nu {)}^n\hat{f}(\nu )$

$x^{n}f(x)$

$i^n\frac{d^n\hat{f}(\nu )}{d{\nu }^n}$

Convolution $(f\ast g)(x)$

$\hat{f}(\nu )\hat{g}(\nu )$

$f(x)g(x)$

$\frac{1}{2\pi }(\hat{f}\ast \hat{g})(\nu )$

Hermitian symmetry. For $f(x)$ purely real

$\hat{f}(-\nu )=\overline{\hat{f}(\nu )}$

For $f(x)$ purely real and even

$f(\nu )$ is purely real even functions.

For $f(x)$ purely real and odd

$f(\nu )$ are purely imaginary and odd.

For $f(x)$ purely imaginary

$\hat{f}(-\nu )=-\overline{\hat{f}(\nu )}$

Complex conjugation $\overline{f(x)}$

$\overline{\hat{f}(-\nu )}$

$f(x)\cos (ax)$

$\frac{\hat{f}(\nu -a)+\hat{f}(\nu +a)}{2}$

$f(x)\sin (ax)$

$\frac{\hat{f}(\nu -a)-\hat{f}(\nu +a)}{2i}$

$e^{-\alpha x^2}$

$\sqrt{\frac{\pi }{\alpha }}\cdot e^{-\frac{{\nu }^2}{4\alpha }}$

$e^{-i\alpha x^2}$

$\sqrt{\frac{\pi }{\alpha }}\cdot e^{i(\frac{{\nu }^2}{4\alpha }-\frac{\pi }{4})}$

$f(x)e^{-\alpha x^2}$

…anything useful?

3 Interpolation

Fourier transforms are useful for cheap interpolations, specifically a magical kind of polynomial basis function.

4 Enveloped Fourier transforms

If you multiply your function by an envelope or tapering function or window or whatever before taking a Fourier transform, then this is no longer strictly FT; but I wonder which features we can relate this back to FTs?

This is heavily studied in the context of time frequency analyses, but I have not found a reference for all the results I need, so I will derive a couple of my own here.

5 References