Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space

Abstract

This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the p-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the “broom topology” type.

Geometrical Theory of Analytic Functions

“…Paper [3] investigates centered polygonal lacunary functions restricted on the unit disk and in the symmetry angle space, which is defined by the symmetry the angles of a given centered polygonal lacunary function. The periodicity of the p-sequences and the existence of a convergent subsequence provided a framework for the decomposition of the centered polygonal lacunary functions.…”

“…Funding: The research presented in [3] was funded by the Concordia College Chemistry Endowment Fund. The research presented in [6] was funded by Universiti Kebangsaan Malaysia, grant number GUP-2019-032.…”