Hermitian function
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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:
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
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
[edit] Motivation
Hermitian functions appear frequently in mathematics and signal processing. As an example, the following statements are important when dealing with Fourier transforms:
- 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.
