weight functions

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1: 37.2 General Orthogonal Polynomials of Two Variables

… ►Then … ►Then necessarily

\left\langle P_{k,n},Q_{k,n}\right\rangle_{W}\neq 0

(

k=0,1,\ldots,n

). … ► … ►

§37.2(v) Product Weight Functions

… ►

§37.2(vii) Rotation Invariant Weight Functions

…

2: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

►A Freud weight is a weight function of the form ►

18.32.1

{w(x)=\exp\left(-Q(x)\right)},

-\infty<x<\infty

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $w(x)$ : weight function and $x$ : real variable
Referenced by:: §18.32
Permalink:: http://dlmf.nist.gov/18.32.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.32 and Ch.18

… ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions

Q(x)

see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). … ►

18.32.2

w(x)={\left|x\right|}^{\alpha}\exp\left(-Q(x)\right),

x\in\mathbb{R}

\alpha>-1

ⓘ

Symbols:: $\in$ : element of, $\exp\NVar{z}$ : exponential function, $\mathbb{R}$ : real line, $w(x)$ : weight function and $x$ : real variable
Referenced by:: §18.32, Erratum (V1.2.0) §18.32
Permalink:: http://dlmf.nist.gov/18.32.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.32 and Ch.18

…

3: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere. …The weight function is invariant under the reflection group

G

. … ►

§37.19(ii) OPs on the Ball for Generalized Weight Functions

… ►

§37.19(iii) OPs on ${\mathbb{R}}^{d}$ for Generalized Weight Functions

►Let

w_{\kappa}

be the weight function (37.19.5). …

4: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Define an inner product … ►

§37.13(i) OPs for a Rotation Invariant Weight Function

… ►For each

k\in\mathbb{N}_{0}

let

p_{n}^{(k)}(x)

denote an OP of degree

n

for the weight function

x^{k}w(x)

[0,\infty)

. The space

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

for the rotation invariant weight function

W_{\circ}

can be orthogonally decomposed as … ►and weight function …

5: 37.10 Other Orthogonal Polynomials of Two Variables

… ►For the weight function … ► … ►The Bernstein–Szegő weight function is defined by …One example of the weight function is …The OPs for these weight functions can be constructed explicitly and they are studied in Delgado et al. (2009). …

6: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

7: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^{2}-y^{2}}$

§37.6 Plane with Weight Function ${\mathrm{e}}^{-x^{2}-y^{2}}$

►On the plane

{\mathbb{R}}^{2}

consider the weight function

{\mathrm{e}}^{-x^{2}-y^{2}}

and the corresponding inner product … ►

§37.6(iii) Differential Equations

… ►

§37.6(iv) Limits

►The explicit basis functions in §37.4 of (bi)orthogonal polynomials on the unit disk for the weight function (37.4.2) all tend after rescaling, as

\alpha\to\infty

, to basis functions given above of OPs on

{\mathbb{R}}^{2}

for the weight function

{\mathrm{e}}^{-x^{2}-y^{2}}

: …

8: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

§37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

… ►bounded by a parabolic arc and a line segment, define the weight function … ►

§37.7(ii) Quadratic Transformations

… ►

§37.7(iii) Differential Equations

… ►The polynomials (37.3.9) and (37.7.16) are related by the quadratic transformations …

9: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials

\{p_{n}(x)\}

n=0,1,\dots

, are orthogonal on

(-1,1)

with respect to three types of weight function:

(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}

(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}

(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}

. …

10: 18.3 Definitions

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$	Constraints
…

► … ►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. …