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:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(-x) = \overline{f(x)}

for all Failed to parse (Missing texvc executable; please see math/README to configure.): x

in the domain of of Failed to parse (Missing texvc executable; please see math/README to configure.): f

.

This definition extends also to functions of two or more variables, e.g., in the case that Failed to parse (Missing texvc executable; please see math/README to configure.): f

is a function of two variables it is Hermitian if

Failed to parse (Missing texvc executable; please see math/README to configure.): f(-x_1, -x_2) = \overline{f(x_1, x_2)}

for all pairs Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1, x_2)

in the domain of Failed to parse (Missing texvc executable; please see math/README to configure.): f

.

From this definition follows immediately that Failed to parse (Missing texvc executable; please see math/README to configure.): f

is a Hermitian function, then

• the real part of Failed to parse (Missing texvc executable; please see math/README to configure.): f

is an even function

• the imaginary part of Failed to parse (Missing texvc executable; please see math/README to configure.): f

is an odd function

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 Failed to parse (Missing texvc executable; please see math/README to configure.): f

is real-valued if and only if the Fourier transform of Failed to parse (Missing texvc executable; please see math/README to configure.): f
is Hermitian.  

• The function Failed to parse (Missing texvc executable; please see math/README to configure.): f

is Hermitian if and only if the Fourier transform of Failed to parse (Missing texvc executable; please see math/README to configure.): f
is real-valued.  

See also

• Even function
• Odd function

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