Poisson transform

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)