The Schauder basis in the solution space of a homogeneous convolution equation
As stated in the title, the author shows that the solution space of a homogeneous convolution equation has a Schauder basis. More precisely, let \(D\) be a convex domain in the complex plane \(C\), and let \(u\) be an analytic functional of \(H^*(D)\) where \(H(D)\) is the space of analytic functions in \(D\) with uniform topology on compact sets of \(D\). Suppose the Laplace transform \( f(z)=(u,\text{exp } \lambda z)\) satisfies certain regularity conditions. Then the solution space \(W(D,u)\) of the convolution equation \((u,f(z+y))=0\) has a Schauder basis. This generalizes an earlier result in \textit{V. V. Napalkov} [Mat. Zametki 43, No. 1, 44-55 (1988; Zbl 0651.46014)] where stronger conditions were required.
scientific article; zbMATH DE number 846459
scientific article