In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, an
orthogonal polynomial sequence is an infinite
sequenceIn 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.-Classes of polynomial sequences:...
of
realIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
polynomials
of one variable
x, in which each
pn has degree
n, and such that any two different polynomials in the sequence are
orthogonalIn mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek , meaning "straight", and , meaning "angle".- Definitions :...
to each other under a particular version of the
L2In mathematics, the L
p spaces are function spaces defined using natural generalizations of p-norms for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to they were first introduced by...
inner product.
The field of orthogonal polynomials developed in the late 19th century from a study of
continued fractionIn mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers. Longer expressions are defined analogously...
s by P. L. Chebyshev and was kept on by
A.A. MarkovAndrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...
and T.J. Stieltjes and by a few other mathematicians. Since then, applications have been developed in many areas of mathematics and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
.
Definition
The theory of orthogonal polynomials includes many definitions of orthogonality. In abstract notation, is convenient to write
when the polynomials
p(x) and
q(x) are orthogonal. A
sequence of orthogonal polynomials, then, is a sequence of polynomials
such that has degree
n and all distinct members of the sequence are orthogonal to each other.
The algebraic and analytic properties of the polynomials depend upon the specific assumptions about the operator . In the classical formulation, the operator is defined in terms of the integral of a weighted product (see below) and happens to be an
inner productIn mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
. Other formulations remove various assumptions, for example in the context of
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s or non-Hermitian operators (see below). Most of the discussion in this article applies to the classical definition.
Classical formulation
Let be an interval in the real line (where and are allowed). This is called the
interval of orthogonality. Let
be a function on the interval, that is strictly positive on the interior , but which may be zero or go to infinity at the end points. Additionally,
W must satisfy the requirement that, for any polynomial , the integral
is finite. Such a
W is called a
weight function.
Given any , , and
W as above, define an operation on pairs of polynomials
f and
g by
This operation is an
inner productIn mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
on the
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of all polynomials. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.
Generalizations
Many alternative theories of orthogonal polynomials have been studied and, along with the classical theory, remain active areas of research. Some aspects of the classical theory generalize when certain assumptions are lifted, and new properties can arise in different contexts.
In some theories the polynomials may act on other algebraic objects such as the complex numbers, matrices, and the unit circle (as a subset of the complex numbers).
Much of the general theory is for operators that satisfy the axioms of an
inner productIn mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
. This includes inner products within a
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
(where the polynomials can be interpreted as an orthogonal basis) and inner products that can be defined as an integral of the form
where μ is a positive
measureIn mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area and volume...
; this in turn includes the classical definition as well as the probabilistic definition (where the measure is a probability measure) and the discrete definition (where the integral is an infinite weighted sum).
The effects of lifting the inner product assumption of positive definiteness have also been studied (e.g. negative weights, discrete coefficients or non-Hermitian operators). In this theory, the terms
system and
sequence of orthogonal polynomials are distinct because pairs of polynomials of the same degree may be orthogonal.
For the remainder of this article the classical definition is assumed.
Standardization
The chosen inner product induces a
normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on polynomials in the usual way:
When making an orthogonal basis, one may be tempted to make an
orthonormal basis, that is,
one in which all basis elements have norm 1. For polynomials, this would often result in
ugly square roots in the coefficients. Instead, polynomials are often scaled in a way
that mathematicians agree on, that makes the coefficients and other formulas simpler. This
is called
standardization. The "classical" polynomials listed below have been standardized,
typically by setting their leading coefficients to some specific quantity, or by setting a
specific value for the polynomial. This standardization has no mathematical significance; it is just a convention. Standardization also involves scaling the weight function in an agreed-upon way.
Denote by the square of the norm of :
The values of for the standardized classical polynomials are listed in the table below. In this notation,
where δ
mn is the
Kronecker deltaIn mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...
.
Example: Legendre polynomials
The simplest classical orthogonal polynomials are the
Legendre polynomialsIn mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields...
, for which the interval of orthogonality is [−1, 1] and the weight function is simply 1:
These are all orthogonal over [−1, 1]; whenever ,
The Legendre polynomials are standardized so that for all
n.
Non-classical example
The simplest non-classical orthogonal polynomials are the monomials
for n≥0 which are orthogonal under the inner product defined by
This inner product cannot be defined in the classical sense as a weighted integral of a product, or even as a measure of a product (otherwise ). However the monomials are orthogonal on the unit circle as a subset of the complex numbers, using the path integral
where denotes complex conjugation. The properties of orthogonal polynomials on the unit circle differ from those of classical orthogonal polynomials (such as the form of the recurrence relations and the distribution of roots) and are related to the theory of
Fourier seriesIn mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
.
General properties of orthogonal polynomial sequences
All orthogonal polynomial sequences have a number of elegant and fascinating properties.
Before proceeding with them:
Lemma 1: Given an orthogonal polynomial sequence , any
nth-degree polynomial
S(
x) can be expanded in terms of . That is, there are
coefficients such that
Proof by
mathematical inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite...
. Choose so that the term of
S(
x) matches that of . Then
is an (
n − 1)th-degree polynomial. Continue downward.
Lemma 2: Given an orthogonal polynomial sequence, each of its polynomials is orthogonal to
any polynomial of strictly lower degree.
Proof: Given
n, any polynomial of degree
n − 1 or lower can be expanded in terms of
. is orthogonal to each of them.
Minimal norm
Each polynomial in an orthogonal sequence has minimal
normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
among all polynomials with the same degree and leading coefficient.
Given
n and any polynomial
p(x) of degree
n with the same leading coefficient can be expanded as
Using orthogonality, the squared norm of
p(x) satisfies
Since the norms are positive, take the square roots of both sides and the result follows.
An interpretation of this result is that orthogonal polynomials are minimal in a generalized
least squaresThe method of least squares is applied to approximate solutions of overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis....
sense. For example, the classical orthogonal polynomials have a minimal weighted mean square value.
Recurrence relations
Any orthogonal sequence has a recurrence formula relating any three consecutive polynomials
in the sequence:
The coefficients
a,
b, and
c depend on
n, as well as the standardization.
We will prove this for fixed
n, and omit the subscripts on
a,
b, and
c.
First, choose
a so that the terms match, so we have
a polynomial of degree
n.
Next, choose
b so that the terms match, so we have
a polynomial of degree
n − 1
Expand the right-hand-side in terms of polynomials in the sequence
Now if , then
But
and
so
Since the inner product is just an integral involving the product:
we have
If , then has degree , so it is orthogonal to ;
hence , which implies for .
Therefore, only can be nonzero, so
Letting , we have
The values of , and can be worked out directly.
Let and be the first and second coefficients of
and be the inner product of with itself:
We have
Existence of real roots
Each polynomial in an orthogonal sequence has all n of its roots real, distinct,
and strictly inside the interval of orthogonality.
Let
m be the number of places where the sign of
Pn changes inside the
interval of orthogonality, and let be those points.
Each of those points is a root of
Pn. By the
fundamental theorem of algebraIn mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
,
m ≤
n.
Now
m might be strictly less than
n if some roots of
Pn are complex, or not inside
the interval of orthogonality, or not distinct. We will show that
m =
n.
Let
This is an
mth-degree polynomial that changes sign at each of the
xj, the same way that
Pn(
x) does.
S(
x)
Pn(
x)
is therefore strictly positive, or strictly negative, everywhere except at
the
xj.
S(
x)
Pn(
x)
W(
x) is also strictly positive
or strictly negative except at the
xj and possibly the end points.
Therefore, , the integral of this, is nonzero.
But, by Lemma 2,
Pn is orthogonal to any polynomial of lower degree,
so the degree of
S must be
n.
Interlacing of roots
The roots of each polynomial lie strictly between the roots of the next higher
polynomial in the sequence.
First, standardize all of the polynomials so that their leading terms are
positive. This will not affect the roots.
Next, a lemma: For all
n and all
x,
Proof by induction. For
n = 0, , ,
and .
Otherwise, the recurrence formula has
with and .
So
So
But by the induction step.
Now if
x is a root of
Pn+1, the lemma tells us that
So and have the same sign.
But must change sign from any root of
Pn+1 to the next. Therefore,
Pn must change sign also, so
Pn must have a root in that interval.
Differential equations leading to orthogonal polynomials
A very important class of orthogonal polynomials arises from a differential equation of the form
where
Q is a given quadratic (at most) polynomial, and
L is a given linear polynomial. The function
f, and the constant λ, are to be found.
This is a
Sturm-LiouvilleIn mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm and Joseph Liouville , is a real second-order linear differential equation of the form...
type of equation. Such equations
generally have singularities in their solution functions f except for particular values
of λ. They can be thought of a eigenvector/eigenvalue problems: Letting
D be the
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .There are certainly reasons not to restrict...
,
, and changing the sign of λ,
the problem is to find the eigenvectors (eigenfunctions) f, and the
corresponding eigenvalues λ, such that f does not have singularities and
D(
f) = λ
f.
The solutions of this differential equation have singularities unless λ takes on
specific values. There is a series of numbers
that lead to a series of polynomial solutions if one
of the following sets of conditions are met:
- Q is actually quadratic, L is linear, Q has two distinct real roots, the root of L lies strictly between the roots of Q, and the leading terms of Q and L have the same sign.
- Q is not actually quadratic, but is linear, L is linear, the roots of Q and L are different, and the leading terms of Q and L have the same sign if the root of L is less than the root of Q, or vice-versa.
- Q is just a nonzero constant, L is linear, and the leading term of L has the opposite sign of Q.
These three cases lead to the
Jacobi-like,
Laguerre-like, and
Hermite-like polynomials, respectively.
In each of these three cases, we have the following:
- The solutions are a series of polynomials , each having degree n, and corresponding to a number .
- The interval of orthogonality is bounded by whatever roots Q has.
- The root of L is inside the interval of orthogonality.
- Letting , the polynomials are orthogonal under the weight function
- W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
- W(x) gives a finite inner product to any polynomials.
- W(x) can be made to be greater than 0 in the interval. (Negate the entire differential equation if necessary so that Q(x) > 0 inside the interval.)
Because of the constant of integration, the quantity
R(
x) is determined only up to an arbitrary positive multiplicative constant. It will be used only in homogeneous differential equations
(where this doesn't matter) and in the definition of the weight function (which can also be
indeterminate.) The tables below will give the "official" values of
R(
x) and
W(
x).
Rodrigues' formula
Under the assumptions of the preceding section,
Pn(
x) is proportional to
This is known as
Rodrigues' formula, after
Olinde RodriguesBenjamin Olinde Rodrigues , more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer....
. It is often written
where the numbers
en depend on the standardization. The standard values of
en will be given in the tables below.
The numbers λn
Under the assumptions of the preceding section, we have
(Since
Q is quadratic and
L is linear, and are constants, so these are just numbers.)
Second form for the differential equation
Let .
Then
Now multiply the differential equation
by
R/
Q, getting
or
This is the standard Sturm-Liouville form for the equation.
Third form for the differential equation
Let .
Then
Now multiply the differential equation
by
S/
Q, getting
or
But , so
or, letting
u =
Sy,
Formulas involving derivatives
Under the assumptions of the preceding section, let
denote the r
th derivative of .
(We put the "r" in brackets to avoid confusion with an exponent.)
is a polynomial of degree
n −
r. Then we have the following:
- (orthogonality) For fixed r, the polynomial sequence are orthogonal, weighted by .
- (generalized Rodrigues'
Benjamin Olinde Rodrigues , more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer....
formula) is proportional to .
- (differential equation) is a solution of , where is the same function as , that is,
- (differential equation, second form) is a solution of
There are also some mixed recurrences. In each of these, the numbers
a,
b, and
c depend on
n
and
r, and are unrelated in the various formulas.
There are an enormous number of other formulas involving orthogonal polynomials
in various ways. Here is a tiny sample of them, relating to the Chebyshev,
associated Laguerre, and Hermite polynomials:
Orthogonality
The differential equation for a particular λ may be written (omitting explicit dependence on x)
multiplying by yields
and reversing the subscripts yields
subtracting and integrating:
but it can be seen that
so that:
If the polynomials
f are such that the term on the left is zero, and for , then the orthogonality relationship will hold:
for .
The classical orthogonal polynomials
The class of polynomials arising from the differential equation described above have many
important applications in such areas as mathematical physics,
interpolation theoryThe Interpolation Theory, also known as the Intercalation Theory or the Antithetic Theory, is a theory that attempts to explain the origin of the alternation of generations in plants...
, the theory of
random matrices,
computer approximationsIn mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
, and many others. All of these polynomial
sequences are equivalent, under scaling and/or shifting of the
domain, and standardizing of the polynomials, to more restricted classes. Those restricted
classes are the "classical orthogonal polynomials".
- Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is [−1, 1], and has Q = 1 − x2. They can then be standardized into the Jacobi polynomials . There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev.
- Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is , and has Q = x. They can then be standardized into the Associated Laguerre polynomials . The plain Laguerre polynomials are a subclass of these.
- Every Hermite-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is , and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials .
Because all polynomial sequences arising from a differential equation in the manner
described above are trivially equivalent to the classical polynomials, the actual classical
polynomials are always used.
Jacobi polynomials
The Jacobi-like polynomials, once they have had their domain shifted and scaled so that
the interval of orthogonality is [−1, 1], still have two parameters to be determined.
They are and in the Jacobi polynomials,
written . We have and
.
Both and are required to be greater than −1.
(This puts the root of L inside the interval of orthogonality.)
When and are not equal, these polynomials
are not symmetrical about
x = 0.
The differential equation
is
Jacobi's equation.
For further details, see
Jacobi polynomialsIn mathematics, Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite:...
.
Gegenbauer polynomials
When one sets the parameters and
in the Jacobi polynomials equal to each other, one obtains the
Gegenbauer or
ultraspherical polynomials. They are
written , and defined as
We have and
.
is required to be greater than −1/2.
(Incidentally, the standardization given in the table below would make no sense for α = 0 and
n ≠ 0, because it would set the polynomials to zero. In that case, the accepted standardization sets instead of the value given in the table.)
Ignoring the above considerations, the parameter is closely related to the derivatives of :
or, more generally:
All the other classical Jacobi-like polynomials (Legendre, etc.) are
special cases of the Gegenbauer polynomials, obtained by choosing a value of
and choosing a standardization.
For further details, see
Gegenbauer polynomialsIn mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. They are named for Leopold Gegenbauer .-Characterizations:...
.
Legendre polynomials
The differential equation is
This is
Legendre's equation.
The second form of the differential equation is:
The recurrence relation is
A mixed recurrence is
Rodrigues' formula is
For further details, see
Legendre polynomialsIn mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields...
.
Associated Legendre polynomials
The Associated Legendre polynomials, denoted
where and are integers with , are defined as
The
m in parentheses (to avoid confusion with an exponent) is a parameter. The
m
in brackets denotes the
mth derivative of the Legendre polynomial.
These "polynomials" are misnamed -- they are not polynomials when
m is odd.
They have a recurrence relation:
For fixed
m, the sequence are orthogonal over [−1, 1], with weight 1.
For given
m, are the solutions of
Chebyshev polynomials
The differential equation is
This is
Chebyshev's equationChebyshev's equation is the second order linear differential equationwhere p is a real constant. The equation is named after Russian mathematician Pafnuty Chebyshev.The solutions are obtained by power series:...
.
The recurrence relation is
Rodrigues' formula is
These polynomials have the property that, in the interval of orthogonality,
(To prove it, use the recurrence formula.)
This means that all their local minima and maxima have values of −1 and +1,
that is, the polynomials are "level". Because of this, expansion of functions
in terms of Chebyshev polynomials is sometimes used for
polynomial
approximationsIn mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
in computer math libraries.
Some authors use versions of these polynomials that have been shifted so that the
interval of orthogonality is [0, 1] or [−2, 2].
There are also
Chebyshev polynomials of the second kind, denoted
We have:
For further details, including the expressions for the first few
polynomials, see
Chebyshev polynomialsIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers...
.
Laguerre polynomials
The most general Laguerre-like polynomials, after the domain has been shifted
and scaled, are the Associated Laguerre polynomials (also called Generalized Laguerre polynomials),
denoted . There is a parameter , which can be any
real number strictly greater than −1. The parameter is put in parentheses to avoid confusion
with an exponent. The plain Laguerre polynomials are simply the
version of these:
The differential equation is
This is
Laguerre's equation.
The second form of the differential equation is
The recurrence relation is
Rodrigues' formula is
The parameter is closely related to the derivatives of :
or, more generally:
Laguerre's equation can be manipulated into a form that is more useful in applications:
is a solution of
This can be further manipulated. When is an integer,
and :
is a solution of
The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials:
This equation arises in quantum mechanics, in the radial part of the solution
of the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
for a one-electron atom.
Physicists often use a definition for the Laguerre polynomials that is larger,
by a factor of , than the definition used here.
For further details, including the expressions for the first few polynomials, see
Laguerre polynomialsIn mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:which is a second-order linear differential equation....
.
Hermite polynomials
The differential equation is
This is
Hermite's equation.
The second form of the differential equation is
The third form is
The recurrence relation is
Rodrigues' formula is
The first few Hermite polynomials are
One can define the
associated Hermite functions
Because the multiplier is proportional to the square root of the weight function, these functions
are orthogonal over with no weight function.
The third form of the differential equation above, for the associated Hermite functions, is
The associated Hermite functions arise in many areas of mathematics and physics.
In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator.
They are also eigenfunctions (with eigenvalue (−
i)
n) of the
continuous Fourier transformIn mathematics, the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. In such applications as signal processing, the domain of the original function is typically time and is accordingly called the time domain...
.
Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of instead of . If the notation
He is used for these Hermite polynomials, and
H for those above, then these may be characterized by
For further details, see
Hermite polynomialsIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum...
.
Constructing orthogonal polynomials by the Gram–Schmidt process
The
Gram–Schmidt processIn mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizng a set of vectors in an inner product space, most commonly the Euclidean space Rn...
is an
algorithmIn mathematics, computing, linguistics, and related subjects, an algorithm is an effective method for solving a problem using a finite sequence of instructions. Algorithms are used for calculation, data processing, and many other fields....
originally taken from
linear algebraLinear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
which removes
linear dependencyIn linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
from a set of given vectors in an
inner product spaceIn mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
.
The inner product as defined on all polynomials allows us to apply the Gram–Schmidt process to an arbitrary set of polynomials. The process removes linear dependencies from the polynomials, yielding sets of orthogonal polynomials. Given various initial polynomial sequences and weighting functions, different orthogonal polynomial sequences can be produced.
We define a
projectionIn linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
operator on the polynomials as:
To apply the algorithm, we define our set of original polynomials and generate a sequence of orthogonal polynomials using:
If an orthonormal sequence is required, a polynomial normalization operation can be defined as:
Care must be taken if the process is implemented on computer as the Gram–Schmidt process is
numerically unstableIn the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
. However, as many computational platforms implement
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s with
arbitrary-precision arithmeticIn computer science, arbitrary-precision arithmetic is a technique whereby calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-point arithmetic found in most ALU hardware, which typically...
the problem can often be easily avoided.
Constructing orthogonal polynomials by using moments
Let
be the
momentsThe concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
of a measure
μ. Then the
polynomial sequenceIn 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.-Classes of polynomial sequences:...
defined by
is a sequence of orthogonal polynomials with respect to the measure
μ. To see this, consider the inner product of
pn(
x) with
xk for any
k <
n. We will see that the value of this inner product is zero.
(The entry-by-entry integration merely says the integral of a
linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
of functions is the same linear combination of the separate integrals. It is a linear combination because only one row contains non-
scalarIn linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
entries.)
Thus
pn(
x) is orthogonal to
xk for all
k <
n. That means this is a sequence of orthogonal polynomials for the measure
μ.
Table of classical orthogonal polynomials
| Name, and conventional symbol |
ChebyshevIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers...
, |
ChebyshevIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers...
(second kind), |
LegendreIn mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields...
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HermiteIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum...
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Associated LaguerreIn mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:which is a second-order linear differential equation....
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LaguerreIn mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:which is a second-order linear differential equation....
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Gegenbauer In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. They are named for Leopold Gegenbauer .-Characterizations:...
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Jacobi In mathematics, Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite:...
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See also
- Polynomial sequences of binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by { 0, 1, 2, 3, ... } in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities...
- Generalized Fourier series
In mathematical analysis, many generalizations of Fourier series have proved to be useful.They are all special cases of decompositions over an orthonormal basis of an inner product space....
- Sheffer sequence
In mathematics, a Sheffer sequence is a polynomial sequence, i.e., a sequence { pn : n = 0, 1, 2, 3, ... } of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isadore M...
- Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence {pn}n = 0, 1, 2, ... satisfying the identity...
- Umbral calculus
In mathematics before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to 'prove' them. These techniques were introduced by John Blissard in 1861 and are sometimes...
- Secondary measure
In mathematics, the secondary measure associated with a measure of positive density when there is one, is a measure of positive density , turning the secondary polynomials associated with the orthogonal polynomials for into an orthogonal system....