Fourier transform

The Fourier transformDlmfMathworldPlanetmath F(s) of a functionMathworldPlanetmath f(t) is defined as follows:


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 continuousMathworldPlanetmath, monotone functions and at every point both one-sided limits exist, the Fourier transform can be inverted:


Sometimes the Fourier transform is also defined without the factor 12π in one direction, but therefore giving the transform into the other direction a factor 12π. 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 equationsMathworldPlanetmath. We denote the Fourier transform of f with respect to t in terms of s by t(f).

  • t(af+bg)=at(f)+bt(g),
    where a and b are constants.

  • t(tf)=ist(f).

  • t(xf)=xt(f).

  • We define the bilateral convolutionMathworldPlanetmath of two functions f1 and f2 as:


    Then the following equation holds:


If f(t) is some signal (maybe a wave) then the frequency domain of f is given as t(f). Rayleigh’s theorem states that then the energy E carried by the signal f given by:


can also be expressed as:


In general we have:


also known as the first Parseval theorem.

Title Fourier transform
Canonical name FourierTransform
Date of creation 2013-03-22 12:34:28
Last modified on 2013-03-22 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