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Zbl 0715.39011
Deeba, E.Y.; Koh, E.L.
The Pexider functional equations in distributions.
(English)
[J] Can. J. Math. 42, No.2, 304-314 (1990). ISSN 0008-414X; ISSN 1496-4279/e

Let $I=(0,+\infty)$ and let D(I) and $D(I\sp 2)$ denote the space of infinitely differentiable complex-valued functions with compact support on I and $I\sp 2$, respectively. The dual of these spaces will be denoted by prime. Let the operators $E\sb 1$, $E\sb 2$ and Q from $D(I\sp 2)$ into D(I) be given by $E\sb 1[\phi](x)=\int\sb{I}\phi (x,y)dy,$ $E\sb 2[\phi](y)=\int\sb{I}\phi (x,y)dx$ and $Q[\phi](x)=\int\sp{+\infty}\sb{- \infty}\phi (x-y,y)dy=\int\sp{+\infty}\sb{-\infty}\phi (y,x-y)dy.$ Their adjoint are $E\sp*\sb 1$, $E\sp*\sb 2$ and $Q\sp*$. If $T,U,V\in D'(I)$ satisfy equation $$(P)\quad Q\sp*[T]=E\sp*\sb 1[U]+E\sp*\sb 2[V],$$ then there exist a,b,c$\in R$ such that $<T,\phi >=\int\sb{I}(cx+a+b)\phi (x)dx,$ $<U,\phi >=\int\sb{I}(cx+a)\phi (x)dx$ and $<V,\phi >=\int\sb{I}(cx+b)\phi (x)dx$ for $\phi\in D(I)$. 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.
[A.Smajdor]
MSC 2000:
*39B52 Functional equations for functions with more general domains
46F10 Operations with distributions (generalized functions)

Keywords: Pexider functional equations; distributions

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