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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: 3.2 Linear Algebra

… ►

Iterative Refinement

… ►Because of rounding errors, the residual vector

\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}

is nonzero as a rule. … ►The $p$ -norm of a vector

\mathbf{x}=[x_{1},\dots,x_{n}]^{\rm T}

is given by … ►The sensitivity of the solution vector

\mathbf{x}

in (3.2.1) to small perturbations in the matrix

\mathbf{A}

and the vector

\mathbf{b}

is measured by the condition number … ►Let

\mathbf{x}^{*}

denote a computed solution of the system (3.2.1), with

\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}^{*}

again denoting the residual. …

3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space

V

is called an inner product space if an inner product

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. …

V

becomes a normed linear vector space. If

\left\|{v}\right\|=1

then

v

is normalized. … ►

The residual spectrum. It consists of all $z\in\mathbb{C}$ for which $z-T$ is injective, but does not have dense range.

… ►If

T

is self-adjoint (bounded or unbounded) then

\sigma(T)

is a closed subset of

\mathbb{R}

and the residual spectrum is empty. …

4: 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 ${\mathrm{e}}^{-\left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}$

…

5: 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

…

6: Bibliography L

… ►

E. Lindelöf (1905) Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Gauthier-Villars, Paris (French).

ⓘ

Notes:: Reprinted by Éditions Jacques Gabay, Sceaux, 1989.
External Links:: ISBN 2-87647-060-8, MathReview, ZentralBlatt
Referenced by:: (25.5.10), (25.5.11), (25.5.12).
Encodings:: BibTeX

… ►

A. E. Lynas-Gray (1993) VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.

ⓘ

Notes:: Single-precision Fortran, minimum accuracy 13D.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

7: 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}

. …

8: 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

. …

9: 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. …

10: 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.
$\mathbf{x},\mathbf{y}$	$(x_{1},\ldots,x_{d}),(y_{1},\ldots,y_{d})\in{\mathbb{R}}^{d}$ .
…
$\left\\|{\mathbf{x}}\right\\|$	$\sqrt{x_{1}^{2}+\cdots+x_{d}^{2}}$ ( $\mathbf{x}\in{\mathbb{R}}^{d}$ ).
…