The Schauder basis in the solution space of a homogeneous convolution equation
Copyright policy )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.
Schauder basis, convex domain, Laplace transform, convolution equations, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, uniform topology on compact sets, Hyperfunctions, analytic functionals, solution space, analytic functional, homogeneous convolution, Topological linear spaces of continuous, differentiable or analytic functions, space of analytic functions
Schauder basis, convex domain, Laplace transform, convolution equations, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, uniform topology on compact sets, Hyperfunctions, analytic functionals, solution space, analytic functional, homogeneous convolution, Topological linear spaces of continuous, differentiable or analytic functions, space of analytic functions
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