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{{Math-h}}

{{Math-h}}

The '''Fourier transformation''' (often called the '''Fourier transform''' or '''Fourier integral''') is an invertible [[integral]] transformation that decomposes a square integrable, [[piecewise continuous]] functions on a [[topological group]] into a linear combination of basis elements.  It can be thought of as the ultimate extension of the [[Fourier series]], in which the interval of periodicity becomes infinitely long and the "coefficients" infinitely close together, becoming a function instead of an infinite series.

The '''Fourier transformation''' (often called the '''Fourier transform''' or '''Fourier integral''') is an invertible [[integral]] transformation that decomposes a square integrable, piecewise continuous functions on a [[topological group]] into a linear combination of basis elements.  It can be thought of as the ultimate extension of the [[Fourier series]], in which the interval of periodicity becomes infinitely long and the "coefficients" infinitely close together, becoming a function instead of an infinite series.

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

The formulas usually favored by mathematicians are the "normalized" form.  Given a function f(k) defined on the entire real line, its Fourier transform g(k) is given by:

The formulas usually favored by mathematicians are the "normalized" form.  Given a function <math>f(t)</math> defined on the entire real line, its Fourier transform <math>g(\omega)</math> is given by:

:<math>g(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ixk}\, dx</math>

:<math>\mathcal{F}[f(t)](\omega) = g(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t}\, dt</math>

The inverse transform, that recovers the original function, is:

The inverse transform, that recovers the original function, is:

:<math>f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(k) e^{ixk}\, dk</math>

:<math>\mathcal{F}^{-1}[g(\omega)](t) = f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(\omega) e^{i \omega t}\, d \omega</math>

The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>.

The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>.

==Discrete Fourier transformation==  

==Discrete Fourier transformation==