Fourier transformation - Revision history

PeterIceHockey: Tidied up the maths

2016-11-23T21:50:23Z

Tidied up the maths

DavidB4-bot: clean up & uniformity

2016-07-13T13:05:48Z

clean up & uniformity

Able806 at 19:33, December 31, 2008

2008-12-31T19:33:40Z

Foxtrot: bold, jargon tag, links, cat

2008-12-06T03:31:57Z

bold, jargon tag, links, cat

Henry8th at 20:02, May 12, 2008

2008-05-12T20:02:56Z

NathanG: nbsp subtraction

2008-03-12T01:00:19Z

nbsp subtraction

CScience at 23:51, May 17, 2007

2007-05-17T23:51:14Z

CScience: Put in actual formulas -- way too elegant to omit.

2007-05-17T23:46:17Z

Put in actual formulas -- way too elegant to omit.

Noodles at 01:58, March 24, 2007

2007-03-24T01:58:34Z

Noodles at 01:58, March 24, 2007

2007-03-24T01:58:15Z

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 {{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.
-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:
-:<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>.
 ==Discrete Fourier transformation==

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-{{Template: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.
@@ Line 18: / Line 18: @@
 Discrete Fourier transformations are defined on discrete topological groups, and the integral is replaced by summation.
-[[category:Calculus]]
+[[Category:Calculus]]

← Older revision		Revision as of 19:33, December 31, 2008
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−	{{~~jargon~~}}	+	{{Template: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.

@@ Line 1: / Line 1: @@
-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.
+{{jargon}}
 Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.
@@ Line 16: / Line 18: @@
 Discrete Fourier transformations are defined on discrete topological groups, and the integral is replaced by summation.
-[[category: mathematics]]
+[[category:Calculus]]

@@ Line 3: / Line 3: @@
 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(t) defined on the entire real line, its Fourier transform g(x) is given by:
+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:
-:<math>g(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-ixt}\, dt</math>
+:<math>g(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ixk}\, dx</math>
 The inverse transform, that recovers the original function, is:
-:<math>f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(x) e^{ixt}\, dx</math>
+:<math>f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(k) e^{ixk}\, dk</math>
 ==Discrete Fourier transformation==

← Older revision		Revision as of 23:51, May 17, 2007
Line 1:		Line 1:
−	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.

← Older revision		Revision as of 01:58, March 24, 2007
Line 1:		Line 1:
−	Fourier transformation is an invertible integral transformation that decompose a square integrable, [[piecewise continuous ~~function~~]] on a [[topological group]] into a linear combination of basis elements.	+	Fourier transformation is an invertible integral transformation that decompose a square integrable, [[piecewise continuous functions ]] on a [[topological group]] into a linear combination of basis elements.

	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.