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:
(where the indicates the complex conjugate) for all in the domain of .
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
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 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.
- If both f and g are Hermitian, then .