weight of

(0.001 seconds)

1—10 of 63 matching pages

1: 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 … ►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). ►Generalized Freud weights have the form … ►For (generalized) Freud weights on a subinterval of

[0,\infty)

2: 37.2 General Orthogonal Polynomials of Two Variables

… ►If

W

is a centrally symmetric weight function, i. … … ►

§37.2(v) Product Weight Functions

… ►

§37.2(vii) Rotation Invariant Weight Functions

… ►There are essentially five admissible PDOs with nonnegative weight function: …

3: 3.5 Quadrature

… ►

Gauss Formula for a Logarithmic Weight Function

… ►

Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.

► ►►

$x_{k}$		$w_{k}$
…

► ►

Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.

► ►►

$x_{k}$		$w_{k}$
…

► … ►

Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.

► ►►

$x_{k}$		$w_{k}$
…

► … ►Then the weights are given by …

4: 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). …

5: 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). …

6: 18.40 Methods of Computation

… ►

A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to

w(x)

, as will be considered in the following paragraphs. … ►The quadrature abscissas

x_{n}

and weights

w_{n}

then follow from the discussion of §3.5(vi). … ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain

w(x)

. … ►The quadrature points and weights can be put to a more direct and efficient use. …

7: 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

can be orthogonally decomposed as … ►and weight function …

8: 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. …

9: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

§37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

►On

{\mathbb{R}}^{d}

consider the weight function

\exp\left(-{\left\|{\mathbf{x}}\right\|}^{2}\right)

and the corresponding inner product … ►Specialization in §37.13(i) of the rotation invariant weight function to

W(\mathbf{x})=\exp\left(-{\left\|{\mathbf{x}}\right\|}^{2}\right)

gives for the corresponding OPs that … ► … ►

§37.17(vi) Hermite Polynomials for Weight Function ${\mathrm{e}}^{-\left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}$

…

10: 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 …

weight of

1—10 of 63 matching pages

1: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

2: 37.2 General Orthogonal Polynomials of Two Variables

§37.2(v) Product Weight Functions

§37.2(vii) Rotation Invariant Weight Functions

3: 3.5 Quadrature

Gauss Formula for a Logarithmic Weight Function

4: 37.10 Other Orthogonal Polynomials of Two Variables

5: 37.19 Other Orthogonal Polynomials of d Variables

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

§37.19(iii) OPs on ℝ d for Generalized Weight Functions

6: 18.40 Methods of Computation

A numerical approach to the recursion coefficients and quadrature abscissas and weights

7: 37.13 General Orthogonal Polynomials of d Variables

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

8: 12.15 Generalized Parabolic Cylinder Functions

9: 37.17 Hermite Polynomials on ℝ d

§37.17 Hermite Polynomials on ℝ d

§37.17(vi) Hermite Polynomials for Weight Function e − ⟨ 𝐀 ⁢ 𝐱 , 𝐱 ⟩

10: 37.7 Parabolic Biangular Region with Weight Function ( 1 − x ) α ⁢ ( x − y 2 ) β

§37.7 Parabolic Biangular Region with Weight Function ( 1 − x ) α ⁢ ( x − y 2 ) β

§37.7(ii) Quadratic Transformations

§37.7(iii) Differential Equations

5: 37.19 Other Orthogonal Polynomials of $d$ Variables

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

7: 37.13 General Orthogonal Polynomials of $d$ Variables

9: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

§37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

§37.17(vi) Hermite Polynomials for Weight Function ${\mathrm{e}}^{-\left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}$

10: 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 $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$