Fourier transforms

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 write 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.

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 \(f(x - a)\,\)

\(e^{- i a \nu} \hat{f}(\nu)\,\)

Frequency shift \(f(x)e^{iax}\,\)

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

Dilation \(f(a x)\,\)

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

Duality \(\hat{f}(x)\,\)

\(2\pi f(-\nu)\,\)

\(\frac{d^n f(x)}{dx^n}\,\)

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

\(x^n f(x)\,\)

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

Convolution \((f * g)(x)\,\)

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

\(f(x) g(x)\,\)

\(\frac{1}{2\pi}\left(\hat{f} * \hat{g}\right)(\nu)\,\)

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 \(\overline{f(x)}\)

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

\(f(x) \cos (a x)\)

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

\(f(x)\sin( ax)\)

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

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

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

\(e^{-i\alpha x^{2}}\)

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

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

…anything useful?

Interpolation

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

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.

References