inner product

SearchAdvancedHelp

(0.001 seconds)

1—10 of 26 matching pages

1: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►

§37.17(vi) Hermite Polynomials for Weight Function ${\mathrm{e}}^{-⟨𝐀𝐱,𝐱⟩}$

… ►

37.17.16 $$H_{𝝂}(𝐱;𝐀)={(-1)}^{|𝝂|}{\mathrm{e}}^{⟨𝐀𝐱,𝐱⟩}{D}_{𝐱}^{𝝂}{\mathrm{e}}^{-⟨𝐀𝐱,𝐱⟩},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors and $𝐱$: vector in $d$-dimensional Euclidean space

Source:

Erdélyi et al. (1953b, (20) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

►

37.17.17 $$G_{𝝂}({𝐀}^{-1}𝐱;𝐀)=H_{𝝂}(𝐱;{𝐀}^{-1})={(-1)}^{|𝝂|}{\mathrm{e}}^{⟨{𝐀}^{-1}𝐱,𝐱⟩}{D}_{𝐱}^{𝝂}{\mathrm{e}}^{-⟨{𝐀}^{-1}𝐱,𝐱⟩}.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors, ${𝐀}^{-1}$: matrix inverse and $𝐱$: vector in $d$-dimensional Euclidean space

Source:

Erdélyi et al. (1953b, (21) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

… ►

37.17.19 $$\sum _{𝝂\in {\mathbb{N}}_0^d}{H}_{𝝂}(𝐱;𝐀)\frac{{𝐲}^{𝝂}}{𝝂!}={\mathrm{e}}^{2⟨𝐀𝐲,𝐱⟩-⟨𝐀𝐲,𝐲⟩}.$$

ⓘ

Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Source:

Erdélyi et al. (1953b, (17) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

►

37.17.20 $$\sum _{𝝂\in {\mathbb{N}}_0^d}{G}_{𝝂}(𝐱;𝐀)\frac{{𝐲}^{𝝂}}{𝝂!}={\mathrm{e}}^{2⟨𝐱,𝐲⟩-⟨{𝐀}^{-1}𝐲,𝐲⟩}.$$

ⓘ

Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, ${𝐀}^{-1}$: matrix inverse, $\mathbb{N}$: set of all positive integers, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Source:

Erdélyi et al. (1953b, (18) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

…

2: 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 $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. …Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►

1.18.3 $$c_n=⟨v,v_n⟩.$$

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

§1.18(i)

Permalink:

http://dlmf.nist.gov/1.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(i), §1.18 and Ch.1

… ►

1.18.12 $$⟨f,g⟩={\int }_a^{b}f(x)\overline{g(x)}d\alpha (x),$$

ⓘ

Defines:

$⟨f,g⟩$: inner product over functions

Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. …

3: 1.2 Elementary Algebra

… ►

1.2.41 $$⟨𝐮,𝐯⟩=\overline{⟨𝐯,𝐮⟩},$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.42 $$⟨\alpha 𝐮,\beta 𝐯⟩=\alpha \overline{\beta }⟨𝐮,𝐯⟩,$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.43 $$⟨𝐯,𝐯⟩=0,$$

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.44 $$⟨𝐮,𝐯⟩=0.$$

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.46 $$\Vert 𝐯\Vert ={\Vert 𝐯\Vert }_2=\sqrt{⟨𝐯,𝐯⟩},$$

ⓘ

Defines:

${\Vert 𝐯\Vert }_2$: $l$² norm and $\Vert 𝐯\Vert$: vector norm ($l$²)

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

…

4: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►On the quadratic domain ${𝕍}^{d+1}$, define the inner product … ►

37.18.8 $$[t(1-t)D_{tt}+2(1-t)⟨𝐱,{\nabla }_{𝐱}⟩D_t+t{\mathrm{\Delta }}_{𝐱}-{⟨𝐱,{\nabla }_{𝐱}⟩}^2+(2\mu +d)D_t-(2\mu +\gamma +d+1)(⟨𝐱,{\nabla }_{𝐱}⟩+t{D}_t)+⟨𝐱,{\nabla }_{𝐱}⟩]u=-n(n+2\mu +\gamma +d)u,$$ $\forall u\in {𝒱}_n({𝕍}^{d+1},W_{\mu ,0,\gamma }),d\ge 2,\mu >-\frac{1}{2},\gamma >-1$,

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain and $W$: weight function in $d$-variables

Proved:

Xu (2020, Theorem 3.2)(proved)

Referenced by:

item 2

Permalink:

http://dlmf.nist.gov/37.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(ii), §37.18(ii), §37.18, Part 37.III and Ch.37

… ►

37.18.10 $$\zeta (𝐱,t,𝐲,s;𝐯)=v_1\sqrt{\frac{1}{2}\left(st+⟨𝐱,𝐲⟩+x_{d+1}{y}_{d+1}\right)}+v_2\sqrt{1-t}\sqrt{1-s}$$

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(ii), §37.18(ii), §37.18, Part 37.III and Ch.37

… ►

37.18.13 $$[t({\mathrm{\Delta }}_{𝐱}+D_{tt})+2⟨𝐱,{\nabla }_{𝐱}⟩D_t-⟨𝐱,{\nabla }_{𝐱}⟩+(2\mu +d-t)D_t]u=-nu,$$ $\forall u\in {𝒱}_n({𝕍}^{d+1},W_{\mu ,0}),\mu >-\frac{1}{2}$,

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain and $W$: weight function in $d$-variables

Proved:

Xu (2020, Theorem 7.3)(proved)

Referenced by:

item 5

Permalink:

http://dlmf.nist.gov/37.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18(iii), §37.18, Part 37.III and Ch.37

… ►

37.18.14 $${𝐑}_n((𝐱,t),(𝐲,s))=\frac{2^{\mu +\frac{d-5}{2}}\mathrm{\Gamma }(\mu +\frac{1}{2})\mathrm{\Gamma }(\mu +\frac{d}{2})}{\pi \mathrm{\Gamma }(\mu )}{\int }_{-1}^1{\int }_0^{\pi }{L}_n^{(2\mu +d-1)}\left(t+s+2\rho \cos \theta \right){\mathrm{e}}^{-\rho \cos \theta }{\left(\rho \sin \theta \right)}^{-\frac{1}{2}(\mu +d-3)}{J}_{\frac{\mu +d-3}{2}}(\rho \sin \theta ){(\sin \theta )}^{2\mu +d-2}{\left(1-u^2\right)}^{\mu -1}d\theta du,$$ $\rho =\sqrt{\frac{1}{2}(ts+⟨𝐱,𝐲⟩+\sqrt{t^2-{\Vert 𝐱\Vert }^2}\sqrt{t^2-{\Vert 𝐱\Vert }^2}u)}$,

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $(a,b)$: open interval, $\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $j$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number and ${𝐑}_n$: reproducing kernel

Proved:

Xu (2021b, Theorem 5.4); Take there in (5.5) with ${\mathrm{\Phi }}_{\kappa }$ given by (2.15) the limit case for ${\kappa }_1=\mathrm{\cdots }={\kappa }_d=0$ and use the formula for $j_{\alpha }(z)$ on p.4 in that paper.(proved)

Referenced by:

§37.18(iii)

Permalink:

http://dlmf.nist.gov/37.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18(iii), §37.18, Part 37.III and Ch.37

…

5: 1.1 Special Notation

… ► ►►►

$x,y$	real variables.
…
$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
…

…

6: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►

37.19.1 $$T_{j}f(𝐱)=\frac{\partial f}{\partial x_j}+\sum _{𝐯\in R_{+}}{\kappa }_{𝐯}\frac{f(𝐱)-f(𝐱{\sigma }_{𝐯})}{⟨𝐱,𝐯⟩}{v}_j,$$

ⓘ

Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\partial x$: partial differential of $x$ , $j$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Sources:

Dunkl (1989, Definition 1.3); Dunkl and Xu (2014, Definition 6.4.2)

Permalink:

http://dlmf.nist.gov/37.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

►where $v_{\mathrm{\ell }}$ is the $\mathrm{\ell }$th component of $𝐯$ and $𝐱{\sigma }_{𝐯}$ denotes the reflection $𝐱{\sigma }_{𝐯}=𝐱-2\frac{⟨𝐱,𝐯⟩}{⟨𝐯,𝐯⟩}𝐯.$ These operators commute; that is, $T_{\mathrm{\ell }}{T}_j=T_j{T}_{\mathrm{\ell }}$ for . …They are orthogonal with respect to the inner product … ►

37.19.4 $$w_{\kappa }(𝐱)=\prod _{𝐯\in R_{+}}{\left|⟨𝐱,𝐯⟩\right|}^{2{\kappa }_{𝐯}}.$$

ⓘ

Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors and $𝐱$: vector in $d$-dimensional Euclidean space

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, Part 37.III and Ch.37

… ►There are many such inner products. …

7: 37.15 Orthogonal Polynomials on the Ball

… ►

37.15.13 $${\left(1-2⟨𝐱,𝐲⟩+{\Vert 𝐲\Vert }^2\right)}^{-\alpha -\frac{1}{2}d}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{2^{|𝝂|}{\left(\alpha +\frac{1}{2}d\right)}_{|𝝂|}}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱),$$ $𝐲\in {ℝ}^d$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proved:

Dunkl and Xu (2014, Proposition 5.2.3); There the polynomial $V_{𝝂}^{(\alpha +\frac{1}{2})}$ is denoted by $V_{\alpha }$.(proved)

Referenced by:

(37.15.14)

Permalink:

http://dlmf.nist.gov/37.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

►

37.15.14 $$C_n^{(\alpha +\frac{1}{2}d)}(⟨𝐱,𝐲⟩)=\sum _{|𝝂|=n}\frac{2^n{\left(\alpha +\frac{1}{2}d\right)}_n}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱),$$ $𝐲\in {𝕊}^{d-1}$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Proof sketch:

Use (18.12.4) and (37.15.13).

Permalink:

http://dlmf.nist.gov/37.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

►

37.15.15 $${\left({(1-⟨𝐱,𝐲⟩)}^2+{\Vert 𝐲\Vert }^2\left(1-{\Vert 𝐱\Vert }^2\right)\right)}^{-\alpha -\frac{1}{2}}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{{(-1)}^{|𝝂|}{\left(2\alpha +1\right)}_{|𝝂|}}{2^{|𝝂|}{\left(\alpha +1\right)}_{|𝝂|}𝝂!}{U}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱){𝐲}^{𝝂},$$ $𝐲\in {ℝ}^d$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proved:

Dunkl and Xu (2014, Proposition 5.2.3)(proved)

Permalink:

http://dlmf.nist.gov/37.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

… ►

37.15.18 $${𝐑}_n^{\alpha }(𝐱,𝐲)={\int }_{-1}^1{Z}_n^{\alpha +\frac{1}{2}d}\left(⟨𝐱,𝐲⟩+t\sqrt{1-{\Vert 𝐱\Vert }^2}\sqrt{1-{\Vert 𝐲\Vert }^2}\right)\frac{{\left(1-t^2\right)}^{\alpha -\frac{1}{2}}}{{\int }_{-1}^1{\left(1-t^2\right)}^{\alpha -\frac{1}{2}}dt}dt,$$ $𝐱,𝐲\in {𝔹}^d$, $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝔹}^d$: unit ball and ${𝐑}_n$: reproducing kernel

Proved:

Dunkl and Xu (2014, Theorem 5.2.8)(proved)

Referenced by:

§37.15(vi), §37.4(i)

Permalink:

http://dlmf.nist.gov/37.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vi), §37.15, Part 37.III and Ch.37

… ►

37.15.19 $${\mathbb{P}}_n^0(𝐱,𝐲)=\frac{2n+d}{d}{\int }_{{𝕊}^{d-1}}{C}_n^{(\frac{1}{2}d)}\left(⟨𝐱,\xi ⟩\right)C_n^{(\frac{1}{2}d)}\left(⟨𝐲,\xi ⟩\right)d\sigma (\xi ),$$ $𝐱,𝐲\in {𝔹}^d$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $(a,b)$: open interval, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${𝔹}^d$: unit ball

Proved:

Dunkl and Xu (2014, Theorem 5.2.9); There replace $\mu =\alpha +\frac{1}{2}$.(proved)

Permalink:

http://dlmf.nist.gov/37.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vi), §37.15, Part 37.III and Ch.37

…

8: 37.2 General Orthogonal Polynomials of Two Variables

… ►Then … ►

37.2.3 $${⟨P_{k,n},P_{j,m}⟩}_W=0,$$ $n\ne m$.

ⓘ

Symbols:

${⟨P_1,P_2⟩}_W$: inner product on the space of polynomials, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables and $W$: weight function in two-variables

Proof sketch:

Consequence of (37.2.2).

Permalink:

http://dlmf.nist.gov/37.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.2(i), §37.2, Part 37.II and Ch.37

… ►Then necessarily ${⟨P_{k,n},Q_{k,n}⟩}_W\ne 0$ ($k=0,1,\mathrm{\dots },n$). … ►

37.2.5 $${⟨P_{k,n},P_{j,n}⟩}_W=h_{k,n}{\delta }_{k,j}$$

ⓘ

Defines:

$h_{k,n}$: positive constant (locally)

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${⟨P_1,P_2⟩}_W$: inner product on the space of polynomials, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables and $W$: weight function in two-variables

Source:

Dunkl and Xu (2014, §2.1)

Permalink:

http://dlmf.nist.gov/37.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.2(i), §37.2, Part 37.II and Ch.37

… ►

37.2.6 $${⟨P_{k,n},P_{j,n}⟩}_W={\delta }_{k,j}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${⟨P_1,P_2⟩}_W$: inner product on the space of polynomials, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables and $W$: weight function in two-variables

Source:

Dunkl and Xu (2014, §2.1)

Permalink:

http://dlmf.nist.gov/37.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.2(i), §37.2, Part 37.II and Ch.37

…

9: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►

1.3.20 $$𝐮=\sum _{i=1}^n{c}_i{𝐚}_i,$$ $c_i=⟨𝐮,{𝐚}_i⟩$.

ⓘ

Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

Erratum (V1.2.0) §1.3

Permalink:

http://dlmf.nist.gov/1.3.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

…

10: 37.11 Spherical Harmonics

… ►Define an (Hermitian) inner product ►

37.11.10 $${⟨f,g⟩}_{{𝕊}^{d-1}}=\frac{1}{{\omega }_d}{\int }_{{𝕊}^{d-1}}f(\xi )\overline{g(\xi )}d\sigma (\xi )$$

ⓘ

Defines:

${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere (locally)

Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${\omega }_d$: surface area of unit sphere ${𝕊}^{d-1}$

Permalink:

http://dlmf.nist.gov/37.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(i), §37.11(i), §37.11, Part 37.III and Ch.37

… ►This inner product is normalized such that ${⟨1_{{𝕊}^{d-1}},1_{{𝕊}^{d-1}}⟩}_{{𝕊}^{d-1}}=1$. … ►

37.11.25 $$𝐊(\xi ,\eta )=k(⟨\xi ,\eta ⟩),$$ $\xi ,\eta \in {𝕊}^{d-1}$ with innerproduct $⟨\xi ,\eta ⟩$,

ⓘ

Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Permalink:

http://dlmf.nist.gov/37.11.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iii), §37.11, Part 37.III and Ch.37

… ►

37.11.27 $${𝐑}_n(\xi ,\eta )=\sum _{j=1}^{N_n^d}{Y}_j(\xi )\overline{Y_j(\eta )}=N_n^d\frac{C_n^{(\frac{1}{2}(d-2))}\left(⟨\xi ,\eta ⟩\right)}{C_n^{(\frac{1}{2}(d-2))}\left(1\right)}=Z_n^{(\frac{1}{2}(d-2))}(⟨\xi ,\eta ⟩),$$ $\xi ,\eta \in {𝕊}^{d-1}$.

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $j$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${𝐑}_n$: reproducing kernel

Proved:

Dai and Xu (2013, (1.2.7), (1.2.8))(proved)

Referenced by:

§37.11(iii), §37.11(iii)

Permalink:

http://dlmf.nist.gov/37.11.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iii), §37.11(iii), §37.11, Part 37.III and Ch.37

…