|
Discrete Orthogonal Polynomials
In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure. Examples include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah polynomials. If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The Racah polynomials give an example of this. Definition Consider a discrete measure \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. 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. Listeratur *{{Citation , last ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 various algebraic properties of spin angular momentum. Elementary Definition The discrete Chebyshev polynomial t^N_n(x) is a polynomial of degree ''n'' in ''x'', for n = 0, 1, 2,\ldots, N -1, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function w(x) = \sum_^ \delta(x-r), with \delta(\cdot) being the Dirac delta function. That is, \int_^ t^N_n(x) t^N_m (x) w(x) \, dx = 0 \quad \text \quad n \ne m . The integral on the left is actually a sum because of the delta function, and we have, \sum_^ t^N_n(r) t^N_m (r) = 0 \quad \text\quad n \ne m. Thus, even though t^N_n(x) is a polynomial in x, only its values at a discrete set of points, x = 0, 1, 2, \ldots, N-1 are of any significance. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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;-;-1/\mu)=(-1)^n n! L_n^\left(-\frac 1 \mu \right), where L are generalized Laguerre polynomials. They satisfy the orthogonality relation :\sum_^\infty \frac C_n(x; \mu)C_m(x; \mu)=\mu^ e^\mu n! \delta_, \quad \mu>0. They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion. See also * Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the ..., a generalization of Charlier polynomials. References * C. V. L. Charlier (1905–1906) ''Über die Darstellung willk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 are (for ''q'' = 2): : \mathcal_0(x; n) = 1, : \mathcal_1(x; n) = -2x + n, : \mathcal_2(x; n) = 2x^2 - 2nx + \binom, : \mathcal_3(x; n) = -\fracx^3 + 2nx^2 - (n^2 - n + \frac)x + \binom. The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind. Definition For any prime power ''q'' and positive integer ''n'', define the Kravchuk polynomial :\mathcal_k(x; n,q) = \mathcal_k(x) = \sum_^(-1)^j (q-1)^ \binom \binom, \quad k=0,1, \ldots, n. Properties The Kravchuk polynomial has the following alternative expressions: :\mathcal_k(x; n,q) = \sum_^(-q)^j (q-1)^ \binom \binom. :\mathcal_k(x; n,q) = \sum_^(-1)^j q^ \binom \binom. Symmetry relations For integers i,k \ge 0, we have that :\begin (q-1)^ \mathcal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meixner Polynomials
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ... by :M_n(x,\beta,\gamma) = \sum_^n (-1)^kk!(x+\beta)_\gamma^ See also * Kravchuk polynomials References * * * * * * * * * * * * * *{{cite journal , first1= Xiang-Sheng , last1=Wang , first2=Roderick , last2=Wong , title= Global asymptotics of the Meixner polynomials , journal = Asymptot. Anal. , year=2011 , volume=75 , number=3–4 , pages=211–231 , doi=10.3233/ASY-2011-1060 , arxiv=1101.4370 Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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,a+s+1;a-b+a,a+c+1;1) for n=0,1,...,N-1 and the parameters a,b,c are restricted to -\frac Orthogonality The dual Hahn polynomials have the orthogonality condit ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by :Q_n(x;\alpha,\beta,N)= _3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\ give a detailed list of their properties. If \alpha = \beta = 0, these polynomials are identical to the discrete Chebyshev polynomials except for a scale factor. Closely related polynomials include the dual Hahn polynomials ''R''''n''(''x'';γ,δ,''N''), the continuous Hahn polynomials ''p''''n''(''x'',' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 given by :p_n(x(x+\gamma+\delta+1)) = _4F_3\left begin -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\ \alpha+1&\gamma+1&\beta+\delta+1\\ \end;1\right Orthogonality :\sum_^N\operatorname_n(x;\alpha,\beta,\gamma,\delta) \operatorname_m(x;\alpha,\beta,\gamma,\delta)\frac \omega_y=h_n\operatorname_, :when \alpha+1=-N, :where \operatorname is the Racah polynomial, :x=y(y+\gamma+\delta+1), :\operatorname_ is the Kronecker delta function and the weight functions are :\omega_y=\frac, :and :h_n=\frac\frac\frac, :(\cdot)_n is the Pochhammer symbol. Rodrigues-type formula :\omega(x;\alpha,\beta,\gamma,\delta)\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)=(\gamma+\delta+1)_n\frac\omega(x;\alpha+n,\beta+n,\gamma+n,\delta), :where \nabla is the backwa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses. Definition and properties A measure \mu defined on the Lebesgue measure, Lebesgue measurable sets of the real line with values in , \infty/math> is said to be discrete if there exists a (possibly finite) sequence of numbers : s_1, s_2, \dots \, such that : \mu(\mathbb R\backslash\)=0. The simplest example of a discrete measure on the real line is the Dirac delta function \delta. One has \delta(\mathbb R\backslash\)=0 and \delta(\)=1. More generally, if s_1, s_2, \dots is a (possibly finite) sequence of real numbers, a_1, a_2, \dots is a sequence of numbers in , \infty/math> of the same length, one can consider the Dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, where the Kronecker delta is a piecewise function of variables and . For example, , whereas . The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above. In linear algebra, the identity matrix has entries equal to the Kronecker delta: I_ = \delta_ where and take the values , and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
