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Hermitian function

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Title: Hermitian function  
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Subject: Charles Hermite, Wannier equation, Autocorrelation, Calculus, Characteristic function (probability theory)
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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) = \overline{f(x)}

(where the overbar indicates the complex conjugate) 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

f(-x_1, -x_2) = \overline{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


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

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