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A potential of the form
 
A potential of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158701.png" /></td> </tr></table>
+
$$
 +
P _  \alpha  (x)  = \
 +
\int\limits _ {\mathbf R  ^ {n} }
 +
G _  \alpha  (x - y) \
 +
d \mu (y),\ \
 +
a > 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158703.png" /> are points in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158705.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158706.png" /> is a Borel measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158707.png" />;
+
where $  x = (x _ {1} \dots x _ {n} ) $,  
 +
$  y = (y _ {1} \dots y _ {n} ) $
 +
are points in the Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $;  
 +
$  d \mu $
 +
is a Borel measure on $  \mathbf R  ^ {n} $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158708.png" /></td> </tr></table>
+
$$
 +
G _  \alpha  (x)  = \
 +
2 ^ {(2 - n - \alpha ) / 2 }
 +
\pi ^ {-n / 2 }
 +
\left [ \Gamma \left (
 +
{
 +
\frac \alpha {2}
 +
} \right )  \right ]  ^ {-1}
 +
K _ {(n - \alpha ) / 2 }
 +
(| x |)  | x | ^ {( \alpha - n) / 2 } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b0158709.png" /></td> </tr></table>
+
$$
 +
| x |  = \left ( \sum _ {i = 1 } ^ { n }  | x _ {i}  ^ {2} | \right )  ^ {1/2} ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587010.png" /> is the modified cylinder function (or Bessel function, cf. [[Cylinder functions|Cylinder functions]]) of the second kind of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587011.png" /> or the Macdonald function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587013.png" /> is called a Bessel kernel.
+
and $  K _  \nu  (z) $
 +
is the modified cylinder function (or Bessel function, cf. [[Cylinder functions|Cylinder functions]]) of the second kind of order $  \nu $
 +
or the Macdonald function of order $  \nu $;  
 +
$  G _  \alpha  (x) $
 +
is called a Bessel kernel.
  
The principal properties of the Bessel kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587014.png" /> are the same as those of the Riesz kernels (cf. [[Riesz potential|Riesz potential]]), viz., they are positive, continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587015.png" />, can be composed
+
The principal properties of the Bessel kernels $  G _  \alpha  (x) $
 +
are the same as those of the Riesz kernels (cf. [[Riesz potential|Riesz potential]]), viz., they are positive, continuous for $  x \neq 0 $,  
 +
can be composed
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\mathbf R  ^ {n} }
 +
G _  \alpha  (x - y)
 +
G _  \beta  (y)  dy  = \
 +
G _ {\alpha + \beta }  (x),
 +
$$
  
but, unlike the Riesz potentials, Bessel potentials are applicable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587017.png" />, since
+
but, unlike the Riesz potentials, Bessel potentials are applicable for all $  \alpha > 0 $,  
 +
since
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587018.png" /></td> </tr></table>
+
$$
 +
G _  \alpha  (x)  \sim \
 +
2 ^ {(1 - n - \alpha ) / 2 }
 +
\pi ^ {(1 - n) / 2 }
 +
\left [ \Gamma \left (
 +
{
 +
\frac \alpha {2}
 +
} \right )  \right ]  ^ {-1}
 +
| x | ^ {( \alpha - n - 1) / 2 }
 +
e ^ {- | x | } ,
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587019.png" />.
+
as $  | x | \rightarrow \infty $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587021.png" /> is a natural number, and the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587022.png" /> is absolutely continuous with square-integrable density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587023.png" />, the Bessel potentials satisfy the identities:
+
If $  \alpha > 2m $,  
 +
where $  m $
 +
is a natural number, and the measure $  d \mu $
 +
is absolutely continuous with square-integrable density $  f(y) \in L _ {2} ( \mathbf R  ^ {2m} ) $,  
 +
the Bessel potentials satisfy the identities:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587024.png" /></td> </tr></table>
+
$$
 +
(1 - \Delta )  ^ {m}
 +
P _  \alpha  (x)  = \
 +
P _ {\alpha - 2m }  (x),
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587025.png" /></td> </tr></table>
+
$$
 +
(1 - \Delta )  ^ {m}
 +
P _ {2m} (x)  = \
 +
f (x),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587026.png" /> is the [[Laplace operator|Laplace operator]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587027.png" />. In other words, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587028.png" /> is a fundamental solution of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587029.png" />.
+
where $  \Delta $
 +
is the [[Laplace operator|Laplace operator]] on $  \mathbf R  ^ {2m} $.  
 +
In other words, the function $  G _ {2m} (x) $
 +
is a fundamental solution of the operator $  (1 - \Delta )  ^ {m} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Aronszajn,  K.T. Smith,  "Theory of Bessel potentials I"  ''Ann. Inst. Fourier (Grenoble)'' , '''11'''  (1961)  pp. 385–475</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Aronszajn,  K.T. Smith,  "Theory of Bessel potentials I"  ''Ann. Inst. Fourier (Grenoble)'' , '''11'''  (1961)  pp. 385–475</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015870/b01587030.png" /> is usually called the modified Bessel function of the third kind.
+
The function $  K _  \nu  (z) $
 +
is usually called the modified Bessel function of the third kind.

Latest revision as of 10:58, 29 May 2020


A potential of the form

$$ P _ \alpha (x) = \ \int\limits _ {\mathbf R ^ {n} } G _ \alpha (x - y) \ d \mu (y),\ \ a > 0, $$

where $ x = (x _ {1} \dots x _ {n} ) $, $ y = (y _ {1} \dots y _ {n} ) $ are points in the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $; $ d \mu $ is a Borel measure on $ \mathbf R ^ {n} $;

$$ G _ \alpha (x) = \ 2 ^ {(2 - n - \alpha ) / 2 } \pi ^ {-n / 2 } \left [ \Gamma \left ( { \frac \alpha {2} } \right ) \right ] ^ {-1} K _ {(n - \alpha ) / 2 } (| x |) | x | ^ {( \alpha - n) / 2 } , $$

$$ | x | = \left ( \sum _ {i = 1 } ^ { n } | x _ {i} ^ {2} | \right ) ^ {1/2} , $$

and $ K _ \nu (z) $ is the modified cylinder function (or Bessel function, cf. Cylinder functions) of the second kind of order $ \nu $ or the Macdonald function of order $ \nu $; $ G _ \alpha (x) $ is called a Bessel kernel.

The principal properties of the Bessel kernels $ G _ \alpha (x) $ are the same as those of the Riesz kernels (cf. Riesz potential), viz., they are positive, continuous for $ x \neq 0 $, can be composed

$$ \int\limits _ {\mathbf R ^ {n} } G _ \alpha (x - y) G _ \beta (y) dy = \ G _ {\alpha + \beta } (x), $$

but, unlike the Riesz potentials, Bessel potentials are applicable for all $ \alpha > 0 $, since

$$ G _ \alpha (x) \sim \ 2 ^ {(1 - n - \alpha ) / 2 } \pi ^ {(1 - n) / 2 } \left [ \Gamma \left ( { \frac \alpha {2} } \right ) \right ] ^ {-1} | x | ^ {( \alpha - n - 1) / 2 } e ^ {- | x | } , $$

as $ | x | \rightarrow \infty $.

If $ \alpha > 2m $, where $ m $ is a natural number, and the measure $ d \mu $ is absolutely continuous with square-integrable density $ f(y) \in L _ {2} ( \mathbf R ^ {2m} ) $, the Bessel potentials satisfy the identities:

$$ (1 - \Delta ) ^ {m} P _ \alpha (x) = \ P _ {\alpha - 2m } (x), $$

and

$$ (1 - \Delta ) ^ {m} P _ {2m} (x) = \ f (x), $$

where $ \Delta $ is the Laplace operator on $ \mathbf R ^ {2m} $. In other words, the function $ G _ {2m} (x) $ is a fundamental solution of the operator $ (1 - \Delta ) ^ {m} $.

References

[1] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[2] M. Aronszajn, K.T. Smith, "Theory of Bessel potentials I" Ann. Inst. Fourier (Grenoble) , 11 (1961) pp. 385–475

Comments

The function $ K _ \nu (z) $ is usually called the modified Bessel function of the third kind.

How to Cite This Entry:
Bessel potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_potential&oldid=13961
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article