arXiv subjects

Natalia Abuzyarova

Publications and source records attributed to Natalia Abuzyarova.

4 recordsLinked to original sources

arXiv2022

Differentiation operator in the Beurling space of ultradifferentiable functions of normal type on an interval

In this paper we study closed subspaces of ultradifferentiable functions which are invariant under the differentiation operator. We propose a version of spectral synthesis which takes into account the presence of non-trivial differentiation invariant subspaces containing no exponential monomials. As an application, we describe the sets of solutions of finite and infinite systems of < > homogeneous convolution equations.

math.CV↗

arXiv2021

Invertibility In the Sense of Ehrenpreis

We consider zero sets of entire functions belonging to the Schwartz algebra. This algebra is defined as the Fourier-Laplace transform image of the space of all distributions compactly supported on the real line. We study the conditions under which given complex sequence forms zero set of some invertible in the sense of Ehrenpreis element of the Schwartz algebra (slowly decreasing function).

math.CV↗

arXiv2016

Spectral synthesis for the differentiation operator in the Schwartz space

We consider the spectral synthesis problem for the differentiation operator D=d/dt in the Schwartz space E(a;b) and the dual problem of local description for closed submodules in a special module of entire functions.

math.CV↗

arXiv2004

Branch point area methods in conformal mapping

The classical estimate of Bieberbach -- that $|a_2|\le2$ for a given univalent function $ϕ(z)=z+a_2z^2+...$ in the class $S$ -- leads to best possible pointwise estimates of the ratio $ϕ''(z)/ϕ'(z)$ for $ϕ\in S$, first obtained by Kœbe 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 $ψ\inΣ$. 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 Kœbe-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.

math.CV↗