Bessel potensiali: Versiyalar orasidagi farq

Matematikada Bessel potentsiali Riesz potentsialiga oʻxshash potentsialdir (Fridrix Vilgelm Bessel nomi bilan atalgan), lekin cheksizlikda yaxshiroq parchalanish xususiyatlariga ega.

Agar s musbat haqiqiy qismga ega kompleks son boʻlsa, s tartibli Bessel potensiali operatori:

${\displaystyle (I-\Delta)^{-s/2}}$

bu yerda Δ – Laplas operatori va kasr quvvati Furye transformlari yordamida aniqlanadi.

Yukava potensiali Bessel potentsiallarining ${\displaystyle s=2}$ 3 oʻlchovli fazoda xususiy holatlaridir

Bessel potentsiali Furye oʻzgarishlariga koʻpaytirish orqali taʼsir qiladi: har bir ${\displaystyle \xi \in \mathbb{R}^d}$uchun:

${\displaystyle \mathcal{F}((I-\Delta)^{-s/2} u) (\xi)= \frac{\mathcal{F}u (\xi)}{(1 + 4 \pi^2 \vert \xi \vert^2)^{s/2}}.}$

${\displaystyle s > 0}$ boʻlganda, Bessel potensiali ${\displaystyle \mathbb{R}^d}$da quyidagicha ifodalanishi mumkin:

${\displaystyle (I - \Delta)^{-s/2} u = G_s \ast u,}$

bu yerda Bessel yadrosi ${\displaystyle G_s}$ ${\displaystyle x \in \mathbb{R}^d \setminus \{0\}}$ uchun integral formula boʻyicha^[1] ifodalanadi

${\displaystyle G_s (x) = \frac{1}{(4 \pi)^{s/2}\Gamma (s/2)} \int_0^\infty \frac{e^{-\frac{\pi \vert x \vert^2}{y}-\frac{y}{4 \pi}}}{y^{1 + \frac{d - s}{2}}}\,\mathrm{d}y.}$

Bu yerda ${\displaystyle \Gamma}$ Gamma funksiyasini bildiradi. Bessel yadrosi^[2] orqali ${\displaystyle x \in \mathbb{R}^d \setminus \{0\}}$ ham ifodalanishi mumkin:

${\displaystyle G_s (x) = \frac{e^{-\vert x \vert}}{(2\pi)^\frac{d-1}{2} 2^\frac{s}{2} \Gamma (\frac{s}{2}) \Gamma (\frac{d - s + 1}{2})} \int_0^\infty e^{-\vert x \vert t} \Big(t + \frac{t^2}{2}\Big)^\frac{d - s - 1}{2} \,\mathrm{d}t.}$

Ushbu oxirgi ifodani oʻzgartirilgan Bessel funksiyasi^[3] nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial oʻz nomini oladi:

${\displaystyle G_s(x)=\frac{1}{2^{(s-2)/2}(2\pi)^{d/2}\Gamma(\frac{s}{2})}K_{(d-s)/2}(\vert x \vert) \vert x \vert^{(s-d)/2}.}$

Asimptotiklar

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Kelib chiqishida birida ${\displaystyle \vert x\vert \to 0}$,^[4]

${\displaystyle G_s (x) = \frac{\Gamma (\frac{d - s}{2})}{2^s \pi^{s/2} \vert x\vert^{d - s}}(1 + o (1)) \quad \text{ if } 0 < s < d,}$

${\displaystyle G_d (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) ,}$

${\displaystyle G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s > d.}$

Xususan, ${\displaystyle 0 < s < d}$ boʻlganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda oʻzini tutadi.

Cheksizlikda, ${\displaystyle \vert x\vert \to \infty}$,^[5]

${\displaystyle G_s (x) = \frac{e^{-\vert x \vert}}{2^\frac{d + s - 1}{2} \pi^\frac{d - 1}{2} \Gamma (\frac{s}{2}) \vert x \vert^\frac{d + 1 - s}{2}}(1 + o (1)).}$

Shuningdek qarang:

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• Riesz potensiali
• Sobolev fazosi
• Shredinger kasr tenglamasi
• Yukava potensiali

Manbalar

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1. ↑ Stein, Elias. Singular integrals and differentiability properties of functions. Princeton University Press, 1970. ISBN 0-691-08079-8. 
2. ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475, (4,2).
3. ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475.
4. ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475, (4,3).
5. ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475-bet.

• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, 250-jild (2nd-nashr), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8