Levinson-type theorem and Dyn'kin problems

Abstract

Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish. The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line. Bibliography: 24 titles.