Free Interpolation in Spaces of Analytic Functions

Abstract

Values taken by an analytic function at different parts of its domain are rigidly interrelated by the classical uniqueness theorem. This is why a function defined on an infinite part Λ of a domain G ⊂ ℂ and taken at random cannot be interpolated, as a rule, by a function analytic in G (f ∈ A(G), for short). In other words, denoting by R Λ the operator of restriction to Λ, we may assert that R Λ(A(G)) is, as a rule, a very special subset of s(Λ), the set of all complex valued functions defined on Λ; R Λ(A(G))resists any description in clear terms. Another way to express this feeling is to say that the equation R Λ(x) = f (with a given f ∈ s(Λ) and “unknown” x ∈ A(G))is strongly overdetermined, it is solvable only under rare circumstances.KeywordsSingular Integral OperatorBlaschke ProductUnconditional BasisInterpolation TheoremBasic FamilyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.