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:

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

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

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

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

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

• the real part of ${\displaystyle f}$ is an even function,
• the imaginary part of ${\displaystyle f}$ 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 ${\displaystyle f}$ is real-valued if and only if the Fourier transform of ${\displaystyle f}$ is Hermitian.
• The function ${\displaystyle f}$ is Hermitian if and only if the Fourier transform of ${\displaystyle f}$ 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 ${\displaystyle f\star g=f*g}$.

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

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

See also

• Even and odd functions