Convolution equations in domains of \(\mathbb{C}^ n\)
The author deals with the study of convolution equations of the form \(\mu*u=f\), where \(\mu \in A'(\mathbb{C}^ n)\) is an analytic functional and \(u \in A(X_ 1)\), \(f \in A (X_ 2)\) are analytic functions in open subsets \(X_ 1\) and \(X_ 2\) of \(\mathbb{C}^ n\). Introducing the concept of \(\mu\)-convexity for carriers of a pair \((X_ 1, X_ 2)\) of open subsets of \(C^ n\), the growth properties of plurisubharmonic and analytic functions in \(\mathbb{C}^ n\), as well as the growth properties of Fourier- Laplace transforms are emphasized. Thus, the author is concerned with the problem of necessary and sufficient conditions for existence of solutions \(u \in A(X_ 1)\) of \(\mu*u=f\) for any \(f \in A (X_ 2)\) and with an approximation problem: to decide if the space \(E_ \mu\), consisting of all linear combinations of exponential solutions of \(\mu*u=0\) in \(\mathbb{C}^ n\), is dense in the space \(N(X_ 1, X_ 2)\) of all solutions \(u \in A(X_ 1)\) of \(\mu*u=0\) in \(X_ 2\).
scientific article; zbMATH DE number 60554
scientific article