In the work we consider a topological module of entire functions, which is the isomorphic image under the Fourier-Laplace transform of Schwarz space ℰ ′ of distributions compactly supported in a finite or infinite interval ( ; ) ⊂ R. We study some properties of closed submodules in module related with local description problem. We also study issues on duality between closed submodules in and subspaces in the space ℰ = ∞ ( ; ) invariant w.r.t. the differentiation.
Abstract. The classical estimate of Bieberbach -that |a2| ≤ 2 for a given univalent function ϕ(z) = z + a2z 2 + . . . in the class S -leads to best possible pointwise estimates of the ratio ϕ ′′ (z)/ϕ ′ (z) for ϕ ∈ S, first obtained by Koebe and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 -by extremality methods -the corresponding best possible pointwise estimates of ψ ′′ (z)/ψ ′ (z) for ψ ∈ Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain the area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that the Koebe-Bieberbach estimate as well as that of Goluzin are both firmly rooted in the area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.
In the work we consider a topological module of entire functions (;), which is the isomorphic image of Fourier-Laplace transform of Schwarz space formed by distributions with compact supports in a finite or infinite segment (;) ⊂ R. We study the conditions ensuring that the principal submodule of module (;) can be uniquely recovered by the zeroes of a generating function.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.