Fourier transform
The Fourier transform^{} $F(s)$ of a function^{} $f(t)$ is defined as follows:
$$F(s)=\frac{1}{\sqrt{2\pi}}{\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{e}^{ist}f(t)\mathit{d}t.$$ 
The Fourier transform exists if $f$ is Lebesgue integrable on the whole real axis.
If $f$ is Lebesgue integrable and can be divided into a finite number of continuous^{}, monotone functions and at every point both onesided limits exist, the Fourier transform can be inverted:
$$f(t)=\frac{1}{\sqrt{2\pi}}{\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{e}^{ist}F(s)\mathit{d}s.$$ 
Sometimes the Fourier transform is also defined without the factor $\frac{1}{\sqrt{2\pi}}$ in one direction, but therefore giving the transform into the other direction a factor $\frac{1}{2\pi}$. So when looking a transform up in a table you should find out how it is defined in that table.
The Fourier transform has some important properties, that can be used when solving differential equations^{}. We denote the Fourier transform of $f$ with respect to $t$ in terms of $s$ by ${\mathcal{F}}_{t}(f)$.

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${\mathcal{F}}_{t}(af+bg)=a{\mathcal{F}}_{t}(f)+b{\mathcal{F}}_{t}(g),$
where $a$ and $b$ are constants. 
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${\mathcal{F}}_{t}\left(\frac{\partial}{\partial t}f\right)=is{\mathcal{F}}_{t}(f).$

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${\mathcal{F}}_{t}\left(\frac{\partial}{\partial x}f\right)=\frac{\partial}{\partial x}{\mathcal{F}}_{t}(f).$

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We define the bilateral convolution^{} of two functions ${f}_{1}$ and ${f}_{2}$ as:
$$({f}_{1}\ast {f}_{2})(t):=\frac{1}{\sqrt{2\pi}}{\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{f}_{1}(\tau ){f}_{2}(t\tau )\mathit{d}\tau .$$ Then the following equation holds:
$${\mathcal{F}}_{t}(({f}_{1}\ast {f}_{2})(t))={\mathcal{F}}_{t}({f}_{1})\cdot {\mathcal{F}}_{t}({f}_{2}).$$
If $f(t)$ is some signal (maybe a wave) then the frequency domain of $f$ is given as ${\mathcal{F}}_{t}(f)$. Rayleigh’s theorem states that then the energy $E$ carried by the signal $f$ given by:
$$E={\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{f(t)}^{2}\mathit{d}t$$ 
can also be expressed as:
$$E={\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{{\mathcal{F}}_{t}(f)(s)}^{2}\mathit{d}s.$$ 
In general we have:
$${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{f(t)}^{2}\mathit{d}t={\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{{\mathcal{F}}_{t}(f)(s)}^{2}\mathit{d}s,$$ 
also known as the first Parseval theorem.
Title  Fourier transform 
Canonical name  FourierTransform 
Date of creation  20130322 12:34:28 
Last modified on  20130322 12:34:28 
Owner  mathwizard (128) 
Last modified by  mathwizard (128) 
Numerical id  17 
Author  mathwizard (128) 
Entry type  Definition 
Classification  msc 42A38 
Related topic  Wavelet 
Related topic  ProgressiveFunction 
Related topic  DiscreteFourierTransform 
Related topic  FourierSeriesInComplexFormAndFourierIntegral 
Related topic  TwoDimensionalFourierTransforms 
Related topic  TableOfGeneralizedFourierAndMeasuredGroupoidTransforms 
Defines  first Parseval theorem 