Poisson transform

The integral transform

(*)

where is a function of bounded variation in every finite interval, and also the transform

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

and let

The following inversion formulas hold for the Poisson transform:

for all , and

almost everywhere.

Let be a convex open acute cone in with vertex at zero and let be the dual cone, that is,

The function

is called the Cauchy kernel of the tube domain . The Poisson transform of a (generalized) function is the convolution (cf. Convolution of functions)

where

is the Poisson kernel of the tube domain (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)