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1: 37 Orthogonal Polynomials of Several Variables

Chapter 37 Orthogonal Polynomials of Several Variables

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2: Yuan Xu

… ►His interest is mostly on higher dimensional problems, such as orthogonal polynomials of several variables, cubature formulas, and mutivariable approximation. ►His well-known book Orthogonal Polynomials of Several Variables (with C. …In 1994, he also published the book Common zeros of polynomials in several variables and higher-dimensional quadrature with Longman Scientific. … ►

37 Orthogonal Polynomials of Several Variables

3: Bille C. Carlson

… ►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. …

4: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Define an inner product … ►

Reproducing and Poisson Kernels

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37.13.5 $${𝐑}_n(𝐱,𝐲)=\sum _{|𝝂|=n}\frac{P_{𝝂}(𝐱)P_{𝝂}(𝐲)}{{⟨P_{𝝂},P_{𝝂}⟩}_W},$$ $𝐱,𝐲\in {ℝ}^d$.

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Symbols:

$\in$: element of, $ℝ$: real line, $n$: nonnegative integer, ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ , $P$: polynomial of $d$-variables, ${⟨f,g⟩}_W$: inner product on $d$-dimensional Euclidean space and $𝐱$: $d$-dimensional vector of real numbers

Referenced by:

§37.14(v), §37.15(vi)

Permalink:

http://dlmf.nist.gov/37.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13 and Ch.37

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5: 1.16 Distributions

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§1.16(vi) Distributions of Several Variables

… ►For a multi-index $𝜶=({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n)$, define ►

1.16.31 $$P(𝐱)=\sum _{𝜶}{c}_{𝜶}{𝐱}^{𝜶}=\sum _{𝜶}{c}_{𝜶}{x}_1^{{\alpha }_1}\mathrm{\cdots }{x}_n^{{\alpha }_n},$$

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Symbols:

$n$: nonnegative integer and $P$: polynomial of several variables

Permalink:

http://dlmf.nist.gov/1.16.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.36 $$⟨ℱ\left(P(𝐃)u\right),\varphi ⟩=⟨P_{-}ℱ\left(u\right),\varphi ⟩=⟨ℱ\left(u\right),P_{-}\varphi ⟩,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), §1.16(viii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E36

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(P(𝐷)u)=P(-𝐱)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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6: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ …

7: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials of several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …In several variables they occur, for , as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989a), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). …

8: 18.37 Classical OP’s in Two or More Variables

… ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …In several variables they occur, for $q=1$, as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). …

9: Bibliography O

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

Encodings:

BibTeX

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10: Bibliography D

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Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-80043-9, MathReview (Marcel de Jeu), ZentralBlatt

Referenced by:

§18.37(ii), §18.37(i).

Encodings:

BibTeX

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