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:

$f^{*}(x)=f(-x)$

(where the $^{*}$ indicates the complex conjugate) for all $x$ in the domain of $f$. 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 $f$ is a function of two variables it is Hermitian if

$f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})$

for all pairs $(x_{1},x_{2})$ in the domain of $f$.

From this definition it follows immediately that: $f$ is a Hermitian function if and only if

• 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 follow 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 f is Hermitian, then $f\star g=f*g$.

Where the $\star$ is cross-correlation, and $*$ is convolution.

• If both f and g are Hermitian, then $f\star g=g\star f$.

See also

• Even and odd functions