# Poisson 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=48224
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