mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, a Hermitian function is a

complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...

with the property that its

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

is equal to the original function with the variable changed in

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

: :

f^*(x) = f(-x)

(where the

^*

indicates the complex conjugate) for all

x

in the domain of

f

. In

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, 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 if :

f^*(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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

* the real part of

f

is an

even function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

, * the imaginary part of

f

is an

odd function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

Motivation

Hermitian functions appear frequently in mathematics, physics, and

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

. 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 In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

f

is Hermitian. * The function

f

is Hermitian if and only if the

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 In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

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

cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...

, and

*

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

. * If both ''f'' and ''g'' are Hermitian, then

f \star g = g \star f

Motivation

See also