Recent zbMATH articles in MSC 30H99https://www.zbmath.org/atom/cc/30H992022-05-16T20:40:13.078697ZWerkzeugThe use of the isometry of function spaces with different numbers of variables in the theory of approximation of functionshttps://www.zbmath.org/1483.300582022-05-16T20:40:13.078697Z"Bushev, D. M."https://www.zbmath.org/authors/?q=ai:bushev.d-m"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Kal'chuk, I. V."https://www.zbmath.org/authors/?q=ai:kalchuk.inna-v"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerimSummary: In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.Invariant spaces of entire functionshttps://www.zbmath.org/1483.460212022-05-16T20:40:13.078697Z"Krivosheev, A. S."https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Krivosheeva, O. A."https://www.zbmath.org/authors/?q=ai:krivosheeva.o-aLet \(D\subset\mathbb{C}\) be a convex domain and let \(H(D)\) be the space of holomorphic functions on \(D\) endowed with the compact open topology. The paper under review deals with the following problem: Let \(W\) be an invariant subspace of the differentiation operator on \(H(D)\). Which conditions ensure that all functions of \(W\) can be extended to entire functions? This problem naturally arises from the problem of expanding convergence domains of exponential series and their special cases, power series and Dirichlet series. \(W\) is assumed to satisfy \textit{spectral synthesis}, i.e., the closure of the span of the eigenvectors of the differentiation operator in \(H(D)\) is the whole \(W\). The following subset of the unit circle \(\mathbb{T}\) is defined,
\[
J(D)= \Bigl\{\omega\in \mathbb T: \ \sup_{z\in D}\text{Re}\,z\omega=+\infty \Bigr\}.
\]
Let \(\Delta:=\{\lambda_k: k\in\mathbb{N}\}\) be the sequence of eigenvalues of the differentiation operator acting on \(W\). Let \(\Xi(\Delta):=\{\overline{\lambda}/|\lambda|: \lambda\in \Delta\} \). The main theorem asserts that the continuation problem has a positive solution when \(\Xi(\Delta)\subset J(D)\). This result was known only under the assumption that \(J(D)\) is open in \(\mathbb{T}\).
Reviewer: Enrique JordÃ¡ (Alicante)