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Fourier transform (Definition)

The Fourier transform $ F(s)$ of a function $ f(t)$ is defined as follows:

$\displaystyle F(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ist}f(t)dt.$

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 one-sided limits exist, the Fourier transform can be inverted:

$\displaystyle f(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{ist}F(s)ds.$

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)$.

  • $ \mathcal{F}_t(af+bg)=a\mathcal{F}_t(f)+b\mathcal{F}_t(g),$
    where $ a$ and $ b$ are constants.
  • $ \mathcal{F}_t\left(\frac{\partial}{\partial t}f\right)=is\mathcal{F}_t(f).$
  • $ \mathcal{F}_t\left(\frac{\partial}{\partial x}f\right)=\frac{\partial}{\partial x}\mathcal{F}_t(f).$
  • We define the bilateral convolution of two functions $ f_1$ and $ f_2$ as:
    $\displaystyle (f_1\ast f_2)(t):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau.$
    Then the following equation holds:
    $\displaystyle \mathcal{F}_t((f_1\ast f_2)(t))=\mathcal{F}_t(f_1)\cdot\mathcal{F}_t(f_2).$
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:
$\displaystyle E=\int_{-\infty}^{\infty}\vert f(t)\vert^2dt$
can also be expressed as:
$\displaystyle E=\int_{-\infty}^{\infty}\vert\mathcal{F}_t(f)(s)\vert^2ds.$
In general we have:
$\displaystyle \int_{-\infty}^{\infty}\vert f(t)\vert^2dt=\int_{-\infty}^{\infty}\vert\mathcal{F}_t(f)(s)\vert^2ds.$



"Fourier transform" is owned by mathwizard.
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See Also: wavelet, progressive function, discrete Fourier transform, Fourier series in complex form and Fourier integral


Attachments:
approximating Fourier integrals with discrete Fourier transforms (Derivation) by stevecheng
Plancherel's theorem (Theorem) by stevecheng
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Cross-references: frequency domain, equation, convolution, terms, differential equations, properties, Transform, factor, one-sided limits, point, monotone functions, continuous, finite, real axis, Lebesgue integrable, function
There are 30 references to this entry.

This is version 13 of Fourier transform, born on 2002-04-07, modified 2006-10-30.
Object id is 2823, canonical name is FourierTransform.
Accessed 25246 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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Discussion
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Better form by dublisk on 2002-06-04 15:39:36
I think it might be better to define the transform as for f, let

F(s) = int( f(x)*e^(-2*pi*i*s*x) dx, x=-oo,oo).

and then correspondingly,


f(x) = int( F(s)*e^(2*pi*s*x) ds, s=-oo,oo).

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