In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, a Hermitian function is a
complex function
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
with the property that its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
is equal to the original function with the variable changed in
sign:
:
(where the
indicates the complex conjugate) for all
in the domain of
. 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
is a function of two variables it is Hermitian if
:
for all pairs
in the domain of
.
From this definition it follows immediately that:
is a Hermitian function
if and only if
* the real part of
is an
even function,
* the imaginary part of
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:
* The function
is real-valued if and only if the
Fourier transform of
is Hermitian.
* The function
is Hermitian if and only if the
Fourier transform of
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
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
of a signal (which is in general complex) to be stored in the same space as the original real signal.
* If ''f'' is Hermitian, then
.
Where the
is
cross-correlation, and
is
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
.
* If both ''f'' and ''g'' are Hermitian, then
.
See also
*
*
Types of functions
Calculus
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