Hermitian function explained

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)} = f( - x)

(where the

^{*} indicates the complex conjugate) for all x in the domain of 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

f is a function of two variables it is Hermitian iff ^{*(x} _{1 ,} x _{2)} = 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

• the real part of

f is an even function,

• the imaginary part of

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

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