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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

►

§1.6(i) Vectors

… ►

Unit Vectors

… ►

Cross Product (or Vector Product)

… ►

§1.6(ii) Vectors: Alternative Notations

…

2: 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 …The OPs of degree

n

with respect to the inner product (37.17.1) form the space

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

. See §37.6 for the case

d=2

. … ►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 ${\mathrm{e}}^{-\left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}$

…

3: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►

Row and Column Vectors

… ►and the corresponding transposed row vector of length

n

is … ►Two vectors

\mathbf{u}

and

\mathbf{v}

are orthogonal if … ►

Vector Norms

…

4: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►Let

\mathcal{V}_{n}\left(\mathbb{V}^{d+1},W\right)

be the space of orthogonal polynomials of degree

n

with respect to the inner product. … ►where

\Delta_{\mathbf{x}}

and

\nabla_{\mathbf{x}}

are the Laplace operator and the gradient vector in the variable

\mathbf{x}

. … ►, the weight function (37.18.2) with

\phi(t)=t

w_{1}(t)=t^{\beta+2\mu-1}{\mathrm{e}}^{-t}

and

w_{2}(\mathbf{x})=W_{\mu-\frac{1}{2}}(\mathbf{x})=(1-{\left\|{\mathbf{x}}% \right\|}^{2})^{\mu-\frac{1}{2}}

(see (37.15.2)). … ►The spaces

\mathcal{V}_{n}\left(\mathbb{V}_{\mathrm{u}}^{d+1},W_{\mu,0}\right)

are eigenspaces of a second order partial differential operator: …where

\Delta_{\mathbf{x}}

and

\nabla_{\mathbf{x}}

are the Laplace operator and the gradient vector in the variable

\mathbf{x}

. …

5: 37.15 Orthogonal Polynomials on the Ball

… ►The OPs of degree

n

with respect to the inner product (37.15.3) form the space

\mathcal{V}_{n}^{d}=\mathcal{V}_{n}^{\alpha}\left(\mathbb{B}^{d}\right)

. See §37.4 for the case

d=2

. The spaces

\mathcal{V}_{n}^{\alpha}\left(\mathbb{B}^{d}\right)

are eigenspaces of a second order partial differential operator, see (37.15.16). … ►The spaces

\mathcal{V}_{n}^{\alpha}\left(\mathbb{B}^{d}\right)

are eigenspaces of a second order partial differential operator: … ►The space

\mathcal{V}_{n}^{\alpha}\left(\mathbf{x};\mathbb{B}^{d}\right)

of OPs on

\mathbb{B}^{d}

of degree

n

\mathbf{x}

can be decomposed as a direct sum of spaces

\mathcal{V}_{m}^{\boldsymbol{{\beta}}}\left(\mathbf{y};\triangle^{d}\right)

of OPs on

\triangle^{d}

of degree

m

\mathbf{y}

(

y_{\ell}=x_{\ell}^{2}

), where

\boldsymbol{{\beta}}

takes different values depending on

\alpha

. …

6: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►where

v_{\ell}

is the

\ell

th component of

\mathbf{v}

and

\mathbf{x}\sigma_{\mathbf{v}}

denotes the reflection

\mathbf{x}\sigma_{\mathbf{v}}=\mathbf{x}-2\frac{\left\langle\mathbf{x},\mathbf% {v}\right\rangle}{\left\langle\mathbf{v},\mathbf{v}\right\rangle}\mathbf{v}.

These operators commute; that is,

T_{\ell}T_{j}=T_{j}T_{\ell}

for

1\leq\ell<j\leq d

. … ►

37.19.4

w_{\kappa}(\mathbf{x})=\prod_{\mathbf{v}\in R_{+}}{\left|\left\langle\mathbf{x% },\mathbf{v}\right\rangle\right|}^{2\kappa_{\mathbf{v}}}.

ⓘ

Symbols:: $\in$ : element of, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space, $w_{\kappa}$ : weight function, $R_{+}$ : set of positive roots, $\mathbf{v}$ : nonnegative function on $R_{+}$ and $\kappa_{\mathbf{v}}$ : something
Sources:: Dunkl (1989, p. 168); Dunkl and Xu (2014, Definition 7.1.1)
Referenced by:: §37.19(ii)
Permalink:: http://dlmf.nist.gov/37.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.19(i), §37.19 and Ch.37

… ►

37.19.6

W_{\kappa,\mu}(\mathbf{x})=w_{\kappa}(\mathbf{x})\left(1-{\left\|{\mathbf{x}}% \right\|}^{2}\right)^{\mu-\frac{1}{2}}

ⓘ

Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm ( $l$ ²), $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space, $W$ : weight function in $d$ -variables on ball, $\kappa$ : real constant and $\mu$ : real constant
Source:: Dunkl and Xu (2014, Definition 8.1.1)
Permalink:: http://dlmf.nist.gov/37.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §37.19(ii), §37.19 and Ch.37

… ►For the radial weight function

{\left\|{\mathbf{x}}\right\|}^{\alpha}(1-{\left\|{\mathbf{x}}\right\|}^{2})^{% \mu-\frac{1}{2}}

(

\mu>-\frac{1}{2}

) on the unit ball, orthogonal polynomials are studied in Xu (2015) and a closed-form formula of the reproducing kernels is established. … ►Orthogonal polynomials for the weight function

w_{\kappa}(\mathbf{x}){\mathrm{e}}^{-{\left\|{\mathbf{x}}\right\|}^{2}}

{\mathbb{R}}^{d}

can be defined explicitly and most of §37.17 can be extended to this more general setting. …

7: 37.1 Notation

… ► ►►►►

$n$	nonnegative integer.
…
$\oplus$	orthogonal (direct) sum of vector spaces.
$\otimes$	tensor product of vector spaces.
…

… ► ►►►►►

$d$	positive integer, usually $\geq 2$ .
…
$\boldsymbol{{1}}$	multi-dimensional vector with all components being unity.
$\mathbf{x},\mathbf{y}$	$(x_{1},\ldots,x_{d}),(y_{1},\ldots,y_{d})\in{\mathbb{R}}^{d}$ .
…
$\left\\|{\mathbf{x}}\right\\|$	$\sqrt{x_{1}^{2}+\cdots+x_{d}^{2}}$ ( $\mathbf{x}\in{\mathbb{R}}^{d}$ ).
…

8: 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

: …

9: 1.1 Special Notation

… ► ►►►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…
$\mathbf{u}$ , $\mathbf{v}$	column vectors.
$\mathbf{E}_{n}$	the space of all $n$ -dimensional vectors.
…

…

10: 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 … ►, for which there are OPs with

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

being eigenspaces of

L

: … ►

In (37.18.14) on the unbounded cone $\mathbb{V}_{\mathrm{u}}^{d}=\{(\mathbf{x},t)\mid\left\|{\mathbf{x}}\right\|% \leq t,\,t\in\mathbb{R}_{+},\,\mathbf{x}\in{\mathbb{R}}^{d-1}\}$ .

… ►Then the corresponding spaces

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

satisfy (37.13.11) with

\lambda_{n}=-n

. …