×

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

Let \(I=(0,+\infty)\) 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 \(E_ 1[\phi](x)=\int_{I}\phi (x,y)dy,\) \(E_ 2[\phi](y)=\int_{I}\phi (x,y)dx\) and \(Q[\phi](x)=\int^{+\infty}_{- \infty}\phi (x-y,y)dy=\int^{+\infty}_{-\infty}\phi (y,x-y)dy.\) Their adjoint are \(E^*_ 1\), \(E^*_ 2\) and \(Q^*\). If \(T,U,V\in D'(I)\) satisfy equation \[ (P)\quad Q^*[T]=E^*_ 1[U]+E^*_ 2[V], \] then there exist a,b,c\(\in R\) such that \(<T,\phi >=\int_{I}(cx+a+b)\phi (x)dx,\) \(<U,\phi >=\int_{I}(cx+a)\phi (x)dx\) and \(<V,\phi >=\int_{I}(cx+b)\phi (x)dx\) 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.
Reviewer: A.Smajdor

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
46F10 Operations with distributions and generalized functions
PDFBibTeX XMLCite
Full Text: DOI