Poisson kernels

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1: 1.15 Summability Methods

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Poisson Kernel

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1.15.13 $$\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta )d\theta =1.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $P(r,\theta )$: Poisson kernel

Permalink:

http://dlmf.nist.gov/1.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.14 $$P(r,\theta )\to 0,$$

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Symbols:

$P(r,\theta )$: Poisson kernel

Permalink:

http://dlmf.nist.gov/1.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.20 $$A(r,\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }P(r,\theta -t)f(t)dt.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $P(r,\theta )$: Poisson kernel and $A(r,\theta )$: Abel mean

Permalink:

http://dlmf.nist.gov/1.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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Poisson Kernel

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2: 37.16 Orthogonal Polynomials on the Hyperoctant

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§37.16(ii) Poisson Kernel

►The Poisson kernel (37.13.6) of ${𝒱}_n^{𝜶}\left({ℝ}_{+}^d\right)$ is given explicitly by ►

37.16.7 $${𝐏}_z^{𝜶}(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n^{𝜶}(𝐱,𝐲)z^n={(1-z)}^{-1}\exp \left(\frac{z(|𝐱|+|𝐲|)}{z-1}\right)\prod _{\mathrm{\ell }=1}^d\frac{\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{{\left(x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z\right)}^{\frac{1}{2}{\alpha }_{\mathrm{\ell }}}}{I}_{{\alpha }_{\mathrm{\ell }}}\left(\frac{2\sqrt{x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z}}{1-z}\right),$$ , $𝐱,𝐲\in {ℝ}_{+}^d$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $|z|$: absolute value (modulus) for complex numbers, $\in$: element of, $\exp z$: exponential function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $ℝ$: real line, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , ${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ , ${ℝ}_{+}^d$: hyperoctant, $x$: real variable and $y$: real variable

Proved:

Dunkl and Xu (2014, Proposition 8.4.8); There set $V_{\kappa }=\mathrm{id}$.(proved)

Permalink:

http://dlmf.nist.gov/37.16.E7

Encodings:

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See also:

Annotations for §37.16(ii), §37.16 and Ch.37

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3: 37.13 General Orthogonal Polynomials of $d$ Variables

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Reproducing and Poisson Kernels

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37.13.5 $${𝐑}_n(𝐱,𝐲)=\sum _{|𝝂|=n}\frac{P_{𝝂}(𝐱)P_{𝝂}(𝐲)}{{⟨P_{𝝂},P_{𝝂}⟩}_W},$$ $𝐱,𝐲\in {ℝ}^d$.

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Symbols:

$\in$: element of, $ℝ$: real line, $n$: nonnegative integer, ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ , $P$: polynomial of $d$-variables, ${⟨f,g⟩}_W$: inner product on $d$-dimensional Euclidean space and $𝐱$: $d$-dimensional vector of real numbers

Referenced by:

§37.14(v), §37.15(vi)

Permalink:

http://dlmf.nist.gov/37.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13 and Ch.37

►Define the Poisson kernel ${𝐏}_z(𝐱,𝐲)$ by ►

37.13.6 $${𝐏}_z(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n(𝐱,𝐲)z^n,$$ ,

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Defines:

${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ (locally)

Symbols:

$|z|$: absolute value (modulus) for complex numbers, ${𝒱}_n^d$: vector space of orthogonal orthogonal polynomials in $d$-variables, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $𝐲$: vector in $d$-dimensional Euclidean space, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ and $𝐱$: $d$-dimensional vector of real numbers

Referenced by:

§37.13, §37.16(ii), §37.17(iv)

Permalink:

http://dlmf.nist.gov/37.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, §37.13 and Ch.37

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4: 18.18 Sums

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§18.18(vii) Poisson Kernels

►See (18.2.41) for the Poisson kernel in case of general OP’s. ►

Laguerre

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Hermite

… ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …

5: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►The Poisson kernel (37.13.6) of ${𝒱}_n\left({ℝ}^d\right)$ is given explicitly by the Mehler formula ►

37.17.11 $${𝐏}_z(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n(𝐱,𝐲)z^n=\frac{1}{{\left(1-z^2\right)}^{\frac{d}{2}}}\exp \left(\frac{2z⟨𝐱,𝐲⟩-z^2({\Vert 𝐱\Vert }^2+{\Vert 𝐲\Vert }^2)}{1-z^2}\right),$$ .

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Symbols:

$|z|$: absolute value (modulus) for complex numbers, $\exp z$: exponential function, $⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ and ${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$

Proof sketch:

Use (18.18.28).

Permalink:

http://dlmf.nist.gov/37.17.E11

Encodings:

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See also:

Annotations for §37.17(iv), §37.17 and Ch.37

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6: 18.2 General Orthogonal Polynomials

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Poisson kernel

►For OP’s $p_n$ with $h_n$ and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …

7: 18.12 Generating Functions

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

8: Bibliography T

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

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On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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External Links:

ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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