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The Fourier transform (ctu Fourier transf. \((Ff) (\vec{\nu}) = \int…
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
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
notes
Eg: convolution
smoothness and convergence
Fourier series \( f(\vec{x}) = \sum_{n=- \infty}^{\infty} f_{k} e^{-2 \pi i kx/ \Delta} \)
Convolution theorem
Parseval-Plancherelle
Schwartz space
Parseval Plancherell
Dirac comb
Poisson-formula
DFT vs FFT, discrete approximation of Fourier transform