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The Pexider functional equations in distributions. (English) Zbl 0715.39011

Let I=(0,+) and let D(I) and D(I 2 ) denote the space of infinitely differentiable complex-valued functions with compact support on I and I 2 , respectively. The dual of these spaces will be denoted by prime. Let the operators E 1 , E 2 and Q from D(I 2 ) into D(I) be given by E 1 [φ](x)= I φ(x,y)dy, E 2 [φ](y)= I φ(x,y)dx and Q[φ](x)= - + φ(x-y,y)dy= - + φ(y,x-y)dy· Their adjoint are E 1 * , E 2 * and Q * . If T,U,VD ' (I) satisfy equation

(P)Q * [T]=E 1 * [U]+E 2 * [V],

then there exist a,b,cR such that <T,φ>= I (cx+a+b)φ(x)dx, <U,φ>= I (cx+a)φ(x)dx and <V,φ>= I (cx+b)φ(x)dx for φD(I). For regular distributions equation (P) reduces to the Pexider equation f(x+y)=g(x)+h(y). Generalizations of another Pexider equation are also considered.

Reviewer: A.Smajdor
39B52Functional equations for functions with more general domains and/or ranges
46F10Operations with distributions (generalized functions)