The Fourier transform

ctu Fourier transf. $(Ff) (\vec{\nu}) = \int f(\vec{x}) e^{-2\pi i \vec{x} \vec{\nu}} d\vec{x}$

Inverse

Riemann-Lebesgue lemma

Fourier transf of f,h

Partial derivatives

Translations !!

Symmetry

Convolution theorem

Dilation: large effect

notes

Eg: convolution

smoothness and convergence

Schwartz space

Parseval Plancherell

Dirac comb

Poisson-formula

Fourier series $ f(\vec{x}) = \sum_{n=- \infty}^{\infty} f_{k} e^{-2 \pi i kx/ \Delta} $

Convolution theorem

Parseval-Plancherelle

Discrete Fourier transform: $ \hat{f_{j}} = \sum_{n=0}^{n-1} f_{k} e^{-2 \pi i k_{j}/ N}$

Inverse

Periodic sequence

Circular translation

Symmetry

Circular convolution

Parseval-Placherelle

DFT vs FFT, discrete approximation of Fourier transform