mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a sequence of discrete

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure. Examples include the

discrete Chebyshev polynomials In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to v ...

Charlier polynomials In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by :C_n(x; \mu)= _2F_0(-n,-x;-; ...

Krawtchouk polynomials Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname ) are discrete orthogonal polynomials associated with the binomial distribution, introduced by . The first few polynomials ar ...

, Meixner polynomials,

dual Hahn polynomials In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice x(s)=s(s+1) and are defined as :w_n^ (s,a,b)=\frac _3F_2(-n,a-s, ...

Hahn polynomials In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spe ...

, and

Racah polynomials In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients. The Racah polynomials were first defined by and are giv ...

. If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The

give an example of this.

Definition

Consider a

discrete measure In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometri ...

\mu

on some set

S=\

with weight function

\omega(x)

. A family of orthogonal polynomials

\

is called discrete if they are orthogonal with respect to

\omega

(resp.

\mu

), i.e., :

\sum\limits_ p_n(x)p_m(x)\omega(x)=\kappa_n\delta_,

where

\delta_

is the

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

Remark

Any discrete measure is of the form :

\mu = \sum_ a_i \delta_

, so one can define a weight function by

\omega(s_i) = a_i

Literature

*{{Citation , last1=Baik , first1=Jinho , last2=Kriecherbauer , first2=T. , last3=McLaughlin , first3=K. T.-R. , last4=Miller , first4=P. D. , title=Discrete orthogonal polynomials. Asymptotics and applications , url=https://books.google.com/books?id=S-GIGgx05kgC , publisher=

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

, series=Annals of Mathematics Studies , isbn=978-0-691-12734-7 , mr=2283089 , year=2007 , volume=164

References

Orthogonal polynomials