On the existence of lacunary power series analytically continuable and bounded outside sectors

Abstract

It is shown that for a set of positive integers Λ, a Müntz-type condition is necessary and sufficient for the existence of a lacunary power series f(z) = ∑_ν∈Λ f _ν/z ^ν which defines a function analytic and bounded in the complement of a sector of given angle and with vertex at the origin. The main tool consists in interpolation of the coefficients f _ν of such functions by certain functions of exponential type in the right half-plane.

Keywords:

• analytic continuation
• lacunary power series
• coefficient functions

AMS Subject Classifications::

• 30B10
• 30B40

Acknowledgement

The research work of V.A. Martirosian has been supported by the Deutsche Forschungsgemeinschaft (DFG grant LU 730/4-1).