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| In [[mathematics]], the Fourier '''sine and cosine transforms''' are forms of the [[Fourier transform|Fourier integral transform]] that do not use [[complex number]]s. They are the forms originally used by [[Joseph Fourier]] and are still preferred in some applications, such as [[signal processing]] or [[statistics]].
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| ==Definition==
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| The '''Fourier sine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^s </math> or <math> {\mathcal F}_s (f) </math>, is
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| :<math> 2 \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>
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| If <math> t </math> means time, then <math> \nu</math> is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.
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| This transform is necessarily an [[odd function]] of frequency, i.e.,
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| :<math> {\hat f}^s(\nu) = - {\hat f}^s(-\nu) </math> for all <math>\nu</math>. | |
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| The numerical factors in the [[Fourier transform]]s are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has <math> L^2 </math> norm of <math> \frac 1 {\sqrt2} </math>.
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| The '''Fourier cosine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^c </math> or <math> {\mathcal F}_c (f) </math>, is
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| :<math> 2 \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math>
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| It is necessarily an [[even function]] of <math>\nu</math>, i.e.,
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| <math> {\hat f}^c(\nu) = {\hat f}^c(-\nu) </math> for all <math>\nu</math>.
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| Some authors<ref>[[Mary L. Boas]], ''[[Mathematical Methods in the Physical Sciences]]'', 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1</ref> only define the cosine transform for [[even function]]s of <math>t </math>, in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, <math> 4 \int\limits_0^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math> Similarly, if <math>f</math> is an [[odd function]], then the cosine transform is zero and the sine transform can be simplified to <math> 4 \int\limits_0^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>
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| ==Fourier inversion==
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| The original function <math> f(t) </math> can be recovered from its transforms under the usual hypotheses, that <math> f </math> and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see [[Fourier inversion theorem]].
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| The inversion formula is<ref>{{cite book|last=[[Poincaré]]|first=Henri|title=Theorie analytique de la propagation de chaleur|year=1895|publisher=G. Carré|location=Paris|pages=pp. 108ff.|url=http://gallica.bnf.fr/ark:/12148/bpt6k5500702f/f115.image}}</ref>
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| :<math> f(t) = \int _0^\infty {\hat f}^c \cos (2\pi \nu t) d\nu + \int _0^\infty {\hat f}^s \sin (2\pi \nu t) d\nu,</math>
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| which has the advantage that all frequencies are positive and all quantities are real. If the numerical factor 2 is left out of the definitions of the transforms, then the inversion formula is usually written as an integral over both negative and positive frequencies.
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| Using the addition formula for [[cosine]], this is sometimes rewritten as
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| :<math> \frac\pi2 (f(x+0)+f(x-0)) = \int _0^\infty \int_{-\infty}^\infty \cos \omega (t-x) f(t) dt d\omega, </math>
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| where <math> f(x+0) </math> denotes the one-sided [[limit]] of <math>f</math> as <math> x </math> approaches zero from above, and
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| <math> f(x-0) </math> denotes the one-sided limit of <math>f</math> as <math> x </math> approaches zero from below.
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| If the original function <math> f</math> is an [[even function]], then the sine transform is zero; if <math> f</math> is an [[odd function]], then the cosine transform is zero. In either case, the inversion formula simplifies.
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| ==Relation with complex exponentials== | |
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| The form of the [[Fourier transform]] used more often today is
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| :<math>
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| \hat f(\nu)
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| = \int\limits_{-\infty}^\infty f(t) e^{-2\pi i\nu t}\,dt.
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| </math>
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| Expanding the [[integrand]] by means of [[Euler's formula]] results in
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| :<math> = \int\limits_{-\infty}^\infty f(t)(\cos\,{2\pi\nu t} - i\,\sin{2\pi\nu t})\,dt,</math> | |
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| which may be written as the [[sum]] of two [[integral]]s
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| :<math> = \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt - i \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t}\,dt,</math>
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| :<math> = \frac 12 {\hat f}^c (\nu) - \frac i2 {\hat f}^s (\nu). </math>
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| ==See also==
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| *[[Discrete cosine transform]]
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| *[[Discrete sine transform]]
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| ==References==
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| * Whittaker, Edmund, and James Watson, ''A Course in Modern Analysis'', Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211
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| <references />
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| [[Category:Integral transforms]]
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| [[Category:Fourier analysis]]
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