Deltahedron: /* References */ expand bibliodata

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References: expand bibliodata

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	~~In [[mathematics]], the '''Riesz potential'''~~ is ~~a [[potential theory\|potential]] named after its discoverer, the [[Hungary\|Hungarian]] [[mathematician]] [[Marcel Riesz]]~~. ~~In a sense, the Riesz potential defines an inverse~~ for ~~a power of the [[Laplace operator]] on Euclidean space~~. ~~They generalize~~ to ~~several variables the [[Riemann–Liouville integral]]s of one variable~~.		Andera is what you can contact her but she never truly favored that title. Distributing production has been his profession for some time. For many years she's been residing in Kentucky but her husband wants them to move. I am really fond of to go to karaoke but I've been using on new issues recently.<br><br>Also visit my blog - clairvoyants - [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ love it],

	~~If 0 < α < ''n'', then the Riesz potential ''~~I~~''<sub>α</sub>''f''~~ of ~~a [[locally integrable function]] ''f'~~' on ~~'''R'''<sup>''n''</sup> is the function defined by~~

	~~{{NumBlk\|:\|<math>(I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{{\mathbb{R}}^n} \frac{f(y)}{\| x - y \|^{n-\alpha}} \, \mathrm{d}y</math>\|{{EquationRef\|1}}}}~~

	~~where the constant is given by~~

	~~:<math>c_\alpha = \pi^{n/2}2^\alpha\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}~~.<~~/math~~>

	~~This [[singular integral]] is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈ [[Lp space\|L<sup>''p''~~<~~/sup~~>('''R'''<sup>''n''</sup>)]] with 1 ≤ ''p'' < ''n''/α. If ''p'' > 1, then the rate of decay of ''f'' and that of ''I''<sub>α</sub>''f'' are related in the form of an inequality (the [[Hardy–Littlewood–Sobolev inequality]])
	~~:<math>\\|I_\alpha f\\|_{p^} \le C_p \\|f\\|_p,\quad p^=\frac{np}{n~~-~~\alpha p}.</math>~~
	~~More generally, the operators ''I''<sub>α</sub> are well~~-~~defined for [[complex number\|complex]] α such that 0 < Re α < ''n''.~~

	~~The Riesz potential can be defined more generally in a [~~[~~distribution (mathematics)\|weak sense]] as the [[convolution]]~~

	:~~<math>I_\alpha f = f*K_\alpha\,<~~/~~math>~~

	~~where ''K''<sub>α<~~/~~sub> is the locally integrable function:~~
	~~:<math>K_\alpha(x) = \frac{1}{c_\alpha}\frac{1}{\|x\|^{n-\alpha}}~~.</~~math>~~
	The Riesz potential can therefore be defined whenever ''f'' is a compactly supported distribution. In this connection, the Riesz potential of a positive [[Borel measure]] μ with [[support (measure theory)\|compact support]] is chiefly of interest in [[potential theory]] because ''I''<sub>α</sub>μ is then a (continuous) [[subharmonic function]] off the support of μ, and is [[lower semicontinuous]] on all of '''R'''<sup>''n''</~~sup>.~~

	~~Consideration of the [[Fourier transform]] reveals that the Riesz potential is a [[Fourier multiplier]]. In fact, one has~~
	~~:<math>\widehat{K_\alpha}(\xi) = \|2\pi\xi\|^{~~-~~\alpha}</math>~~
	~~and so, by the [[convolution theorem]],~~
	~~:<math>\widehat{I_\alpha f}(\xi) = \|2\pi\xi\|^{~~-~~\alpha} \hat{f}(\xi).</math>~~

	~~The Riesz potentials satisfy the following [[semigroup]] property on, for instance, rapidly decreasing continuous functions~~
	~~:<math>I_\alpha I_\beta = I_{\alpha+\beta}\ </math>~~
	~~provided~~
	~~:<math>0 < \operatorname{Re\,} \alpha, \operatorname{Re\,} \beta < n,\quad 0 < \operatorname{Re\,} (\alpha+\beta) < n.</math>~~
	~~Furthermore, if 2 < Re α <''n'', then~~
	~~:<math>\Delta I_{\alpha+2} =~~ -~~I_\alpha.\ </math>~~
	~~One also has, for this class of functions,~~
	~~:<math>\lim_{\alpha\to 0^+} (I^\alpha f)(x) = f(x).</math>~~

	~~==See also==~~
	* [[Bessel potential]]
	* [[Fractional integration]]
	* [[Sobolev space]]
	* [[Fractional Schrödinger equation]]

	~~==References==~~
	*{{Citation \| last1=Landkof \| first1=N. S. \| title=Foundations of modern potential theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| id={{MathSciNet \| id = 0350027}} \| year=1972}}
	*{{Citation \| last1=Riesz \| first1=Marcel \| author1-~~link=Marcel Riesz \| title=L'intégrale de Riemann~~-~~Liouville et le problème de Cauchy \| id={{MathSciNet \| id = 0030102}} \| year=1949 \| journal=[[Acta Mathematica]] \| issn=0001~~-~~5962 \| volume=81 \| pages=1–223 \| doi=10.1007/BF02395016}}.~~
	* {{springer\|last=Solomentsev\|first=E.D.\|id=R/~~r082270\|title=Riesz potential}}~~
	* {{citation\|first=Elias\|last=Stein\|authorlink=Elias Stein\|title=Singular integrals and differentiability properties of functions\|publisher=[[Princeton University Press]]~~\|location=Princeton~~, ~~NJ\|year=1970\|isbn=0-691-08079-8}}~~

	~~[[Category:Fractional calculus]]~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Potential theory]]~~
	~~[[Category:Singular integrals]]~~

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2011-07-27T01:17:44Z

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In [[mathematics]], the '''Riesz potential''' is a [[potential theory|potential]] named after its discoverer, the [[Hungary|Hungarian]] [[mathematician]] [[Marcel Riesz]]. In a sense, the Riesz potential defines an inverse for a power of the [[Laplace operator]] on Euclidean space. They generalize to several variables the [[Riemann–Liouville integral]]s of one variable.

If 0 < α < ''n'', then the Riesz potential ''I''α''f'' of a [[locally integrable function]] ''f'' on '''R'''''n'' is the function defined by

{{NumBlk|:|<math>(I_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{{\mathbb{R}}^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y</math>|{{EquationRef|1}}}}

where the constant is given by

:<math>c_\alpha = \pi^{n/2}2^\alpha\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}.</math>

This [[singular integral]] is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈ [[Lp space|L''p''('''R'''''n'')]] with 1 ≤ ''p'' < ''n''/α. If ''p'' > 1, then the rate of decay of ''f'' and that of ''I''α''f'' are related in the form of an inequality (the [[Hardy–Littlewood–Sobolev inequality]])
:<math>\|I_\alpha f\|_{p^*} \le C_p \|f\|_p,\quad p^*=\frac{np}{n-\alpha p}.</math>
More generally, the operators ''I''α are well-defined for [[complex number|complex]] α such that 0 < Re α < ''n''.

The Riesz potential can be defined more generally in a [[distribution (mathematics)|weak sense]] as the [[convolution]]

:<math>I_\alpha f = f*K_\alpha\,</math>

where ''K''α is the locally integrable function:
:<math>K_\alpha(x) = \frac{1}{c_\alpha}\frac{1}{|x|^{n-\alpha}}.</math>
The Riesz potential can therefore be defined whenever ''f'' is a compactly supported distribution. In this connection, the Riesz potential of a positive [[Borel measure]] μ with [[support (measure theory)|compact support]] is chiefly of interest in [[potential theory]] because ''I''αμ is then a (continuous) [[subharmonic function]] off the support of μ, and is [[lower semicontinuous]] on all of '''R'''''n''.

Consideration of the [[Fourier transform]] reveals that the Riesz potential is a [[Fourier multiplier]]. In fact, one has
:<math>\widehat{K_\alpha}(\xi) = |2\pi\xi|^{-\alpha}</math>
and so, by the [[convolution theorem]],
:<math>\widehat{I_\alpha f}(\xi) = |2\pi\xi|^{-\alpha} \hat{f}(\xi).</math>

The Riesz potentials satisfy the following [[semigroup]] property on, for instance, rapidly decreasing continuous functions
:<math>I_\alpha I_\beta = I_{\alpha+\beta}\ </math>
provided
:<math>0 < \operatorname{Re\,} \alpha, \operatorname{Re\,} \beta < n,\quad 0 < \operatorname{Re\,} (\alpha+\beta) < n.</math>
Furthermore, if 2 < Re α <''n'', then
:<math>\Delta I_{\alpha+2} = -I_\alpha.\ </math>
One also has, for this class of functions,
:<math>\lim_{\alpha\to 0^+} (I^\alpha f)(x) = f(x).</math>

==See also==
* [[Bessel potential]]
* [[Fractional integration]]
* [[Sobolev space]]
* [[Fractional Schrödinger equation]]

==References==
*{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | id={{MathSciNet | id = 0350027}} | year=1972}}
*{{Citation | last1=Riesz | first1=Marcel | author1-link=Marcel Riesz | title=L'intégrale de Riemann-Liouville et le problème de Cauchy | id={{MathSciNet | id = 0030102}} | year=1949 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=81 | pages=1–223 | doi=10.1007/BF02395016}}.
* {{springer|last=Solomentsev|first=E.D.|id=R/r082270|title=Riesz potential}}
* {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1970|isbn=0-691-08079-8}}

[[Category:Fractional calculus]]
[[Category:Partial differential equations]]
[[Category:Potential theory]]
[[Category:Singular integrals]]

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Deltahedron: /* References */ expand bibliodata

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