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

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 <i>p</i>-sequences of the cen...

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Main Authors: Leah K. Mork, Keith Sullivan, Darin J. Ulness
Format: Article
Language:English
Published: MDPI AG 2020-04-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/8/4/568
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author Leah K. Mork
Keith Sullivan
Darin J. Ulness
author_facet Leah K. Mork
Keith Sullivan
Darin J. Ulness
author_sort Leah K. Mork
collection DOAJ
description 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 <i>p</i>-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the <i>p</i>-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.
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spelling doaj.art-286452890a314b899794b7e3903ec61b2023-11-19T21:22:45ZengMDPI AGMathematics2227-73902020-04-018456810.3390/math8040568Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle SpaceLeah K. Mork0Keith Sullivan1Darin J. Ulness2Department of Mathematics, Concordia College, Moorhead, MN 56562, USADepartment of Mathematics, Concordia College, Moorhead, MN 56562, USADepartment of Chemistry, Concordia College, Moorhead, MN 56562, USAThis 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 <i>p</i>-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the <i>p</i>-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.https://www.mdpi.com/2227-7390/8/4/568lacunary functiongap functioncentered polygonal numbersnatural boundarysingularitiesbroom topology
spellingShingle Leah K. Mork
Keith Sullivan
Darin J. Ulness
Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
Mathematics
lacunary function
gap function
centered polygonal numbers
natural boundary
singularities
broom topology
title Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
title_full Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
title_fullStr Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
title_full_unstemmed Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
title_short Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space
title_sort taming the natural boundary of centered polygonal lacunary functions restriction to the symmetry angle space
topic lacunary function
gap function
centered polygonal numbers
natural boundary
singularities
broom topology
url https://www.mdpi.com/2227-7390/8/4/568
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