Bessel potensiali sahifasining manbasini koʻrish

[[Matematika|Matematikada]] '''Bessel potentsiali''' '''Riesz potentsialiga''' oʻxshash potentsialdir ([[Bessel Fridrix Vilgelm|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:

: <math>(I-\Delta)^{-s/2}</math>

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

[[Yukava potensiali]] Bessel potentsiallarining <math>s=2</math> 3 oʻlchovli fazoda xususiy holatlaridir

== Furye fazosida koʻrinishi ==
Bessel potentsiali Furye oʻzgarishlariga koʻpaytirish orqali taʼsir qiladi: har bir <math>\xi \in \mathbb{R}^d</math>uchun:

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

== Integral koʻrinishlari ==
<math>s > 0</math> boʻlganda, Bessel potensiali <math>\mathbb{R}^d</math>da quyidagicha ifodalanishi mumkin:

: <math>(I - \Delta)^{-s/2} u = G_s \ast u,</math>

bu yerda Bessel yadrosi <math>G_s</math> <math>x \in \mathbb{R}^d \setminus \{0\} </math> uchun integral formula boʻyicha<ref>{{Kitob manbasi|last=Stein|first=Elias|title=Singular integrals and differentiability properties of functions|url=https://archive.org/details/singularintegral0000stei|date=1970|publisher=Princeton University Press|isbn=0-691-08079-8}}</ref> ifodalanadi

: <math>
 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.
</math>

Bu yerda <math>\Gamma</math> [[Gamma-funksiya|Gamma funksiyasini]] bildiradi. Bessel yadrosi<ref>{{jurnal manbasi|last=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|at=385–475, (4,2)}}</ref> orqali <math>x \in \mathbb{R}^d \setminus \{0\} </math> ham ifodalanishi mumkin:

: <math>
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.
</math>

Ushbu oxirgi ifodani oʻzgartirilgan Bessel funksiyasi<ref>{{jurnal manbasi|last=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|at=385–475}}</ref> nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial oʻz nomini oladi:

: <math>
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}.
</math>

== Asimptotiklar ==
Kelib chiqishida birida <math>\vert x\vert \to 0 </math>,<ref>{{jurnal manbasi|last=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|at=385–475, (4,3)}}</ref>

: <math>
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,
</math>
: <math>
G_d (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) ,
</math>
: <math>
G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s > d.
</math>

Xususan, <math>0 < s < d</math> boʻlganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda oʻzini tutadi.

Cheksizlikda, <math>\vert x\vert \to \infty </math>,<ref>{{jurnal manbasi|last=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|pages=385–475}}</ref>

: <math>
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)).
</math>

== Shuningdek qarang: ==

* [[Riesz potensiali]]
* [[Sobolev fazosi]]
* [[Shredinger kasr tenglamasi]]
* [[Yukava potensiali]]

== Manbalar ==
{{Manbalar}}

*
* {{Citation|last=Grafakos|first1=Loukas|title=Modern Fourier analysis|publisher=[[Springer-Verlag]]|place=Berlin, New York|edition=2nd|series=[[Graduate Texts in Mathematics]]|isbn=978-0-387-09433-5|doi=10.1007/978-0-387-09434-2|mr=2463316|year=2009|volume=250}}
* {{Citation|first1=Elias|last=Stein|author-link=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=[[Princeton University Press]]|place=Princeton, NJ|year=1970|isbn=0-691-08079-8|url=https://archive.org/details/singularintegral0000stei}}
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