In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hermite polynomials are a classical
orthogonal polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in ...
.
The polynomials arise in:
*
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
as
Hermitian wavelets for
wavelet transform analysis
*
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
, such as the
Edgeworth series, as well as in connection with
Brownian motion;
*
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, as an example of an
Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity
:\frac p_n(x) = np_(x),
and in which p_0(x) is a non-zero constant.
Among the most notable Appell sequences besides th ...
, obeying the
umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
;
*
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
as
Gaussian quadrature;
*
physics
Physics is the natural science that studies matter, its 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 which ...
, where they give rise to the
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s of the
quantum harmonic oscillator; and they also occur in some cases of the
heat equation (when the term
is present);
*
systems theory in connection with nonlinear operations on
Gaussian noise.
*
random matrix theory in
Gaussian ensembles.
Hermite polynomials were defined by
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
in 1810, though in scarcely recognizable form, and studied in detail by
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebysh ...
in 1859. Chebyshev's work was overlooked, and they were named later after
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
...
, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials in his later 1865 publications.
Definition
Like the other
classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:
* The "probabilist's Hermite polynomials" are given by
* while the "physicist's Hermite polynomials" are given by
These equations have the form of a
Rodrigues' formula and can also be written as,
The two definitions are not exactly identical; each is a rescaling of the other:
These are Hermite polynomial sequences of different variances; see the material on variances below.
The notation and is that used in the standard references.
The polynomials are sometimes denoted by , especially in probability theory, because
is the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
for the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
0 and
standard deviation 1.
* The first eleven probabilist's Hermite polynomials are:
* The first eleven physicist's Hermite polynomials are:
Properties
The th-order Hermite polynomial is a polynomial of degree . The probabilist's version has leading coefficient 1, while the physicist's version has leading coefficient .
Symmetry
From the Rodrigues formulae given above, we can see that and are
even or odd functions depending on :
Orthogonality
and are th-degree polynomials for . These
polynomials are orthogonal with respect to the ''weight function'' (
measure)
or
i.e., we have
Furthermore,
or
where
is the
Kronecker delta.
The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.
Completeness
The Hermite polynomials (probabilist's or physicist's) form an
orthogonal basis of the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of functions satisfying
in which the inner product is given by the integral
including the
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
weight function defined in the preceding section
An orthogonal basis for is a
''complete'' orthogonal system. For an orthogonal system, ''completeness'' is equivalent to the fact that the 0 function is the only function orthogonal to ''all'' functions in the system.
Since the
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if satisfies
for every , then .
One possible way to do this is to appreciate that the
entire function
vanishes identically. The fact then that for every real means that the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of is 0, hence is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay.
In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the
Completeness relation below).
An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for consists in introducing Hermite ''functions'' (see below), and in saying that the Hermite functions are an orthonormal basis for .
Hermite's differential equation
The probabilist's Hermite polynomials are solutions of the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
where is a constant. Imposing the boundary condition that should be polynomially bounded at infinity, the equation has solutions only if is a non-negative integer, and the solution is uniquely given by
, where
denotes a constant.
Rewriting the differential equation as an
eigenvalue problem
the Hermite polynomials
may be understood as
eigenfunctions of the differential operator