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| In [[mathematical analysis]], a '''Hermitian function''' is a [[complex number|complex]] [[function (mathematics)|function]] with the property that its [[complex conjugate]] is equal to the original function with the variable changed in [[sign (mathematics)|sign]]:
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| :<math>f(-x) = \overline{f(x)}</math>
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| for all <math>x</math> in the domain of <math>f</math>.
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| This definition extends also to functions of two or more variables, e.g., in the case that <math>f</math> is a function of two variables it is Hermitian if
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| :<math>f(-x_1, -x_2) = \overline{f(x_1, x_2)}</math> | |
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| for all pairs <math>(x_1, x_2)</math> in the domain of <math>f</math>.
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| From this definition it follows immediately that, if <math>f</math> is a Hermitian function, then
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| * the real part of <math>f</math> is an [[even function]]
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| * the imaginary part of <math>f</math> is an [[odd function]]
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| == Motivation ==
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| Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:
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| * The function <math>f</math> is real-valued if and only if the [[Fourier transform]] of <math>f</math> is Hermitian.
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| * The function <math>f</math> is Hermitian if and only if the [[Fourier transform]] of <math>f</math> is real-valued.
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| Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the [[discrete Fourier transform]] of a signal (which is in general complex) to be stored in the same space as the original real signal.
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| * If ''f'' is Hermitian, then <math>f \star g = f*g</math>
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| Where the <math> \star </math> is [[cross-correlation]], and <math> * </math> is [[convolution]].
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| * If both ''f'' and ''g'' are Hermitian, then <math>f \star g = g \star f</math>, which in general is not true.
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| <!--------An example wanted for these two statements above! ------->
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| == See also ==
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| * [[Even and odd functions]]
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| [[Category:Types of functions]]
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| [[Category:Calculus]]
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| {{mathanalysis-stub}}
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The individual who wrote the article is known as Jayson Hirano and he totally digs that title. Alaska is where I've always been residing. He is an information officer. Doing ballet is some thing she would never give up.
My blog; psychic readers; http://cartoonkorea.com/ce002/1093612,