orthogonal polynomials

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1: 18.37 Classical OP’s in Two or More Variables

… ►

18.37.2

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)R^{(% \alpha)}_{j,\ell}\left(x-\mathrm{i}y\right)\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm% {d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►The following three conditions, taken together, determine

R^{(\alpha)}_{m,n}\left(z\right)

uniquely: … ►

18.37.4

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm{d}x\,\mathrm{d}y=0,

j=1,2,\dots,\min(m,n)

;

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

►

18.37.5

R^{(\alpha)}_{m,n}\left(1\right)=1.

ⓘ

Symbols:: $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►

18.37.8

\iint\limits_{0<y<x<1}P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)P^{\alpha,% \beta,\gamma}_{j,\ell}\left(x,y\right)\*(1-x)^{\alpha}(x-y)^{\beta}y^{\gamma}% \,\mathrm{d}x\,\mathrm{d}y=0,

m\neq j

and/or

n\neq\ell

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\left(\NVar{x}% ,\NVar{y}\right)$ : triangle polynomial, $y$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

…

2: 37 Orthogonal Polynomials of Several Variables

Chapter 37 Orthogonal Polynomials of Several Variables

…

3: René F. Swarttouw

… ►Swarttouw is mainly a teacher of mathematics and has published a few papers on special functions and orthogonal polynomials. He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their $q$ -Analogues) Hypergeometric Orthogonal Polynomials and Their

q

-Analogues. … ►

18 Orthogonal Polynomials

…

4: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►The OPs of degree

n

with respect to the inner product (37.3.2) form the space

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

. ►The spaces

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

are eigenspaces of a second order partial differential operator, see (37.3.14). … ►They form an orthogonal basis of

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

: … ►two further orthogonal bases of

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

: …

5: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►The OPs of degree

n

with respect to the inner product (37.16.3) form the space

\mathcal{V}_{n}^{d}=\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}\left(\mathbb{R}_{+% }^{d}\right)

. See §37.5 for the case

d=2

. ►The spaces

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}\left(\mathbb{R}_{+}^{d}\right)

are eigenspaces of a second order partial differential operator: … ►Obviously, an orthogonal basis of

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}\left(\mathbb{R}_{+}^{d}\right)

consisting of products of Laguerre polynomials is given by … ►The basis functions (37.16.5) and (37.16.6) of the space

\mathcal{V}_{n}^{\boldsymbol{{\alpha}}}\left(\mathbb{R}_{+}^{d}\right)

are limits of the basis functions (37.14.7) of the space

\mathcal{V}_{n}^{\boldsymbol{{\alpha}},\beta}\left(\triangle^{d}\right)

\mathcal{V}_{n}^{\beta,\boldsymbol{{\alpha}}}\left(\triangle^{d}\right)

, after rescaling, as

\beta\to\infty

: …

6: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

…

7: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Let

\mathcal{V}_{n}^{d}

denote the space of OPs of degree

n

d

variables, i. … ►Similarly to the case

d=2

in §37.2(iii), define the reproducing kernel

\mathbf{R}_{n}(\mathbf{x},\mathbf{y})

(

\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{d}

) of

\mathcal{V}_{n}^{d}

as a polynomial in

\mathbf{y}

belonging to

\mathcal{V}_{n}^{d}

\mathbf{x}

is fixed, and such that … ►The space

\mathcal{V}_{n}^{d}

for the rotation invariant weight function

W_{\circ}

can be orthogonally decomposed as … ►, for which there are OPs with

\mathcal{V}_{n}^{d}

being eigenspaces of

L

: … ►Then the corresponding spaces

\mathcal{V}_{n}^{d}

satisfy (37.13.11) with

\lambda_{n}=-n

. …

8: Yuan Xu

… ►Xu has published numerous papers on analysis including approximation theory, harmonic analysis, orthogonal polynomials, numerical analysis, and special functions. 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. … ►Xu has also served several terms as Secretary of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. … ►

37 Orthogonal Polynomials of Several Variables

9: 37.5 Quarter Plane with Weight Function $x^{\alpha}y^{\beta}{\mathrm{e}}^{-x-y}$

… ►The OPs of degree

n

with respect to the inner product (37.5.3) form the space

\mathcal{V}_{n}^{\alpha,\beta}

. The spaces

\mathcal{V}_{n}^{\alpha,\beta}

are eigenspaces of a second order partial differential operator, see (37.5.8). … ►These polynomials form a second orthogonal basis of

\mathcal{V}_{n}^{\alpha,\beta}

, … ►The spaces

\mathcal{V}_{n}^{\alpha,\beta}

of OPs on

\mathbb{R}_{+}^{2}

are eigenspaces of a second order partial differential operator as follows. … ►Special bases of

\mathcal{V}_{n}^{\alpha,\beta}

can be obtained as joint eigenfunctions of the PDO in (37.5.8) and of another PDO commuting with the first one. …

10: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

orthogonal polynomials

1—10 of 138 matching pages

1: 18.37 Classical OP’s in Two or More Variables

2: 37 Orthogonal Polynomials of Several Variables

Chapter 37 Orthogonal Polynomials of Several Variables

3: René F. Swarttouw

4: 37.3 Triangular Region with Weight Function x α ⁢ y β ⁢ ( 1 − x − y ) γ

5: 37.16 Orthogonal Polynomials on the Hyperoctant

6: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

7: 37.13 General Orthogonal Polynomials of d Variables

8: Yuan Xu

9: 37.5 Quarter Plane with Weight Function x α ⁢ y β ⁢ e − x − y

10: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

4: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

7: 37.13 General Orthogonal Polynomials of $d$ Variables

9: 37.5 Quarter Plane with Weight Function $x^{\alpha}y^{\beta}{\mathrm{e}}^{-x-y}$