Hermitian function

In mathematical analysis, a '''Hermitian function''' is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:

:<math>f^*(x) = f(-x)</math>

(where the <math>^*</math> indicates the complex conjugate) for all <math>x</math> in the domain of <math>f</math>. In physics, this property is referred to as PT symmetry.

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

:<math>f^*(x_1, x_2) = f(-x_1, -x_2)</math>

for all pairs <math>(x_1, x_2)</math> in the domain of <math>f</math>.

From this definition it follows immediately that: <math>f</math> is a Hermitian function if and only if

* the real part of <math>f</math> is an even function, * the imaginary part of <math>f</math> is an odd function.

== Motivation == Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:{{Citation needed|reason=Proof is not entirely trivial.|date=March 2018}}

* The function <math>f</math> is real-valued if and only if the Fourier transform of <math>f</math> is Hermitian. * The function <math>f</math> is Hermitian if and only if the Fourier transform of <math>f</math> is real-valued. 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. Informally, only half of the fourier transform of a real signal is needed to lossessly represent it in frequency domain.

For the magnitude spectra (obtained from DFT), the axis of symmetry is around the Nyquist point; one half is the mirror image of the other.

* If ''f'' is Hermitian, then <math>f \star g = f*g</math>.

Where the <math> \star </math> is cross-correlation, and <math> * </math> is convolution.

* If both ''f'' and ''g'' are Hermitian, then <math>f \star g = g \star f</math>.

== See also ==

* {{annotated link|Complex conjugate}} * {{annotated link|Even and odd functions}}

Category:Types of functions Category:Calculus