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[parent] table of Fourier transforms (Feature)

Below are tables of Fourier transforms; one lists some of the common properties, and the other lists some common examples.

Properties

Original Transformed comment derivation
$ af(t)+bg(t)$ $ a\mathcal{F}\lbrace f(t)\rbrace +b\mathcal{F}\lbrace g(t)\rbrace $ linearity  
$ f(t)*g(t)$ $ \mathcal{F}\lbrace f(t)\rbrace \mathcal{F}\lbrace g(t)\rbrace $ convolution property  
$ f(t+\alpha)$ % latex2html id marker 288 $ F(s)\exp(-i \alpha s)$ time shift, where $ F(s)=\mathcal{F}\lbrace f(t)\rbrace $  
$ f'(t)$ $ is \mathcal{F}\lbrace f(t)\rbrace $ differentiation  
$ \overline{f(t)}$ $ \overline{F(-s)}$ conjugation, where $ F(s)=\mathcal{F}\lbrace f(t)\rbrace $  
$ f(\alpha t)$ $ \displaystyle{\frac{1}{\vert\alpha\vert}F(\frac{s}{\alpha})}$ scaling, where $ F(s)=\mathcal{F}\lbrace f(t)\rbrace $ with $ \alpha\ne 0$  

Examples

$ f(t)$ $ \mathcal{F}\lbrace f(t)\rbrace $ conditions explanation derivation
$ \delta(t)$ $ 1$   Dirac delta function  
$ 1$ $ 2\pi \delta(s)$      
$ e^{i a t}$ $ 2\pi \delta(s - \alpha)$ $ a\in \mathbb{R}$    
$ \cos(at)$ $ \pi (\delta(s+a) + \delta(s-a))$ $ a\in \mathbb{R}$    
$ \sin(at)$ $ i \pi (\delta(s+a) - \delta(s-a))$ $ a\in \mathbb{R}$    



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"table of Fourier transforms" is owned by CWoo. [ full author list (3) ]
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Other names:  Groupoid Transforms
Also defines:  Fourier-Stieltjes generalization of FT

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Cross-references: Dirac delta function, scaling, conjugation, differentiation, convolution, derivation, properties
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This is version 4 of table of Fourier transforms, born on 2008-06-09, modified 2008-07-03.
Object id is 10689, canonical name is TableOfFourierTransforms.
Accessed 627 times total.

Classification:
AMS MSC: 42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
Pending Errata and Addenda
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