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The [[Integral transform|integral transform]]
 
The [[Integral transform|integral transform]]
  
<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/p/p073/p073390/p0733901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f ( x)  =
 +
\frac{1} \pi
 +
 
 +
\int\limits _ {- \infty } ^ { {+ }  \infty }
 +
 
 +
\frac{1}{1 + ( x - t )  ^ {2} }
 +
  d \alpha ( t)
 +
$$
 +
 
 +
where  $  \alpha ( t) $
 +
is a [[Function of bounded variation|function of bounded variation]] in every finite interval, and also the transform
 +
 
 +
$$
 +
f ( x)  =
 +
\frac{1} \pi
 +
 
 +
\int\limits _ {- \infty } ^  \infty 
 +
 
 +
\frac{\phi ( t) }{1 + ( x - t ) ^ {2} }
 +
\
 +
d t
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p0733902.png" /> is a [[Function of bounded variation|function of bounded variation]] in every finite interval, and also the transform
+
which results from (*) if  $  \alpha ( t) $
 +
is an absolutely-continuous function (cf. [[Absolute continuity|Absolute continuity]]). Let
  
<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/p/p073/p073390/p0733903.png" /></td> </tr></table>
+
$$
 +
\widehat{g}  ( x)  = -  
 +
\frac{1} \pi
  
which results from (*) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p0733904.png" /> is an absolutely-continuous function (cf. [[Absolute continuity|Absolute continuity]]). Let
+
\int\limits _ { 0 } ^  \infty 
  
<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/p/p073/p073390/p0733905.png" /></td> </tr></table>
+
\frac{g ( x + u ) - 2 g ( x) + g( x - u ) }{u  ^ {2} }
 +
  d u
 +
$$
  
 
and let
 
and let
  
<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/p/p073/p073390/p0733906.png" /></td> </tr></table>
+
$$
 +
T _ {t} g ( x)  = \
 +
\sum _ { k= } 0 ^  \infty  ( - 1 )  ^ {k}
 +
 
 +
\frac{t  ^ {2k} }{( 2 k ) ! }
 +
 
 +
g ^ {( 2 k ) } ( x) +
 +
$$
  
<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/p/p073/p073390/p0733907.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ { k= } 0 ^  \infty  ( - 1 )  ^ {k}
 +
\frac{t  ^ {2k+} 1 }{( 2 k + 1 ) ! }
 +
\widehat{g}  {}  ^ {(} 2k) ( x) .
 +
$$
  
 
The following inversion formulas hold for the Poisson transform:
 
The following inversion formulas hold for the Poisson transform:
  
<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/p/p073/p073390/p0733908.png" /></td> </tr></table>
+
$$
  
<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/p/p073/p073390/p0733909.png" /></td> </tr></table>
+
\frac{\alpha ( x + 0 ) + \alpha ( x - 0 ) }{2}
 +
-
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339010.png" />, and
+
\frac{\alpha ( + 0 ) + \alpha ( - 0 ) }{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/p/p073/p073390/p07339011.png" /></td> </tr></table>
+
$$
 +
= \
 +
\lim\limits _ {t \uparrow 1 }  \int\limits _ { 0 } ^ { x }  T _ {t} f ( u)  d u
 +
$$
 +
 
 +
for all  $  x $,
 +
and
 +
 
 +
$$
 +
\phi ( x)  = \
 +
\lim\limits _ {t \uparrow 1 } \
 +
T _ {t} f ( x)
 +
$$
  
 
almost everywhere.
 
almost everywhere.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339012.png" /> be a convex open acute cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339013.png" /> with vertex at zero and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339014.png" /> be the dual cone, that is,
+
Let $  C $
 +
be a convex open acute cone in $  \mathbf R  ^ {n} $
 +
with vertex at zero and let $  C  ^ {*} $
 +
be the dual cone, that is,
  
<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/p/p073/p073390/p07339015.png" /></td> </tr></table>
+
$$
 +
C  ^ {*}  = \{  \xi  : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} \geq  0 \textrm{ for  all  }  x \in C } \}
 +
.
 +
$$
  
 
The function
 
The function
  
<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/p/p073/p073390/p07339016.png" /></td> </tr></table>
+
$$
 +
{\mathcal K} _ {C} ( z)  = \
 +
\int\limits _ { C }
 +
e ^ {i ( z _ {1} \xi _ {1} + \dots + z _ {n} \xi _ {n} ) }  d \xi
 +
$$
  
is called the Cauchy kernel of the tube domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339017.png" />. The Poisson transform of a (generalized) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339018.png" /> is the convolution (cf. [[Convolution of functions|Convolution of functions]])
+
is called the Cauchy kernel of the tube domain $  T  ^ {C} = \{ {z = x + i y } : {x \in \mathbf R  ^ {n} ,  y \in C } \} $.  
 +
The Poisson transform of a (generalized) function $  f $
 +
is the convolution (cf. [[Convolution of functions|Convolution of functions]])
  
<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/p/p073/p073390/p07339019.png" /></td> </tr></table>
+
$$
 +
f \star {\mathcal P} _ {C} ( x , y ) ,\ \
 +
( x , y ) \in T  ^ {C} ,
 +
$$
  
 
where
 
where
  
<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/p/p073/p073390/p07339020.png" /></td> </tr></table>
+
$$
 +
{\mathcal P} _ {C} ( x , y )  = \
 +
 
 +
\frac{| {\mathcal K} _ {C} ( x + i y ) |  ^ {2} }{( 2 \pi )  ^ {n} {\mathcal K} _ {C} ( i y ) }
 +
 
 +
$$
  
is the Poisson kernel of the tube domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073390/p07339021.png" /> (see [[#References|[2]]]).
+
is the Poisson kernel of the tube domain $  T  ^ {C} $(
 +
see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Pollard,  "The Poisson transform"  ''Trans. Amer. Math. Soc.'' , '''78''' :  2  (1955)  pp. 541–550</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Pollard,  "The Poisson transform"  ''Trans. Amer. Math. Soc.'' , '''78''' :  2  (1955)  pp. 541–550</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1977)  (Translated from Russian)</TD></TR></table>

Revision as of 08:06, 6 June 2020


The integral transform

$$ \tag{* } f ( x) = \frac{1} \pi \int\limits _ {- \infty } ^ { {+ } \infty } \frac{1}{1 + ( x - t ) ^ {2} } d \alpha ( t) $$

where $ \alpha ( t) $ is a function of bounded variation in every finite interval, and also the transform

$$ f ( x) = \frac{1} \pi \int\limits _ {- \infty } ^ \infty \frac{\phi ( t) }{1 + ( x - t ) ^ {2} } \ d t $$

which results from (*) if $ \alpha ( t) $ is an absolutely-continuous function (cf. Absolute continuity). Let

$$ \widehat{g} ( x) = - \frac{1} \pi \int\limits _ { 0 } ^ \infty \frac{g ( x + u ) - 2 g ( x) + g( x - u ) }{u ^ {2} } d u $$

and let

$$ T _ {t} g ( x) = \ \sum _ { k= } 0 ^ \infty ( - 1 ) ^ {k} \frac{t ^ {2k} }{( 2 k ) ! } g ^ {( 2 k ) } ( x) + $$

$$ + \sum _ { k= } 0 ^ \infty ( - 1 ) ^ {k} \frac{t ^ {2k+} 1 }{( 2 k + 1 ) ! } \widehat{g} {} ^ {(} 2k) ( x) . $$

The following inversion formulas hold for the Poisson transform:

$$ \frac{\alpha ( x + 0 ) + \alpha ( x - 0 ) }{2} - \frac{\alpha ( + 0 ) + \alpha ( - 0 ) }{2\ } = $$

$$ = \ \lim\limits _ {t \uparrow 1 } \int\limits _ { 0 } ^ { x } T _ {t} f ( u) d u $$

for all $ x $, and

$$ \phi ( x) = \ \lim\limits _ {t \uparrow 1 } \ T _ {t} f ( x) $$

almost everywhere.

Let $ C $ be a convex open acute cone in $ \mathbf R ^ {n} $ with vertex at zero and let $ C ^ {*} $ be the dual cone, that is,

$$ C ^ {*} = \{ \xi : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} \geq 0 \textrm{ for all } x \in C } \} . $$

The function

$$ {\mathcal K} _ {C} ( z) = \ \int\limits _ { C } e ^ {i ( z _ {1} \xi _ {1} + \dots + z _ {n} \xi _ {n} ) } d \xi $$

is called the Cauchy kernel of the tube domain $ T ^ {C} = \{ {z = x + i y } : {x \in \mathbf R ^ {n} , y \in C } \} $. The Poisson transform of a (generalized) function $ f $ is the convolution (cf. Convolution of functions)

$$ f \star {\mathcal P} _ {C} ( x , y ) ,\ \ ( x , y ) \in T ^ {C} , $$

where

$$ {\mathcal P} _ {C} ( x , y ) = \ \frac{| {\mathcal K} _ {C} ( x + i y ) | ^ {2} }{( 2 \pi ) ^ {n} {\mathcal K} _ {C} ( i y ) } $$

is the Poisson kernel of the tube domain $ T ^ {C} $( see [2]).

References

[1] H. Pollard, "The Poisson transform" Trans. Amer. Math. Soc. , 78 : 2 (1955) pp. 541–550
[2] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1977) (Translated from Russian)
How to Cite This Entry:
Poisson transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_transform&oldid=14304
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article