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:

$f(-x) = \overline{f(x)}$

for all x in the domain of f.

This definition extends also to functions of two or more variables, e.g., in the case that f is a function of two variables it is Hermitian if

$f(-x_1, -x_2) = \overline{f(x_1, x_2)}$

for all pairs (x₁,x₂) in the domain of f.

From this definition it follows immediately that, if f is a Hermitian function, then

• the real part of f is an even function
• the imaginary part of f is an odd function

Motivation

Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follows from basic properties of the Fourier transform:

• The function f is real-valued if and only if the Fourier transform of f is Hermitian.

• The function f is Hermitian if and only if the Fourier transform of f 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.

• If either f or g is Hermitian, then $f \star g = f*g$

Where the $\star$ is correlation, and * is convolution. Because convolution is commutative we can infer also that:

• If either f or g is Hermitian, then $f \star g = g \star f$, which in general is not true.

See also

• Even and odd functions