Whitney Decomposition, Embedding Theorems, and Interpolation in Weighted Spaces of Analytic Functions

Shamoyan, F. A. , Tasoeva, E. V.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 1.

Abstract:
According to the classical Whitney theorem, each open set on the plane can be decomposed as a union of special squares whose interiors do not intersect. In the paper, using the properties of Whitney squares, a new concept is introduced. For each center \(a_k\) of the Whitney square, there is a point \(a_k^*\in \mathbb{C}\setminus G\) such that the distance to the boundary of the open set \(G\) is between two constants, regardless of \(k\). In particular, a necessary and sufficient condition for a sequence \((z_k)_1^{\infty}\subset G\) under which the operator \(R(f)=(f(z_1),f(z_2),\ldots,f(z_n),\ldots)\) maps generalized Nevanlinna's flat classes in a domain \(G\) of a complex plane in \(l^p.\)

Keywords: Nevanlinna class, interpolation, Witny decomposition, Berman space.

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