Determination of a Universal Series

Abstract

The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connected domains, when universal Taylor series exist, we can construct a sequence of specific rational functions converging to a universal function, provided the boundary is good enough. The solution uses an infinite denumerable procedure and a finite number of steps is not sufficient. However we solve a Runge's type problem in a finite number of steps.

Simultaneous Universal Padé Approximation

“…We mention that the above construction is a modification of the construction of universal Taylor series without use of Baire's theorem ( [3] [8] [14] [13]). Definition 5.3.…”