On the path properties of a lacunary power series

Jensen, Gerd; Pommerenke, Ch.; Ramı́rez, Jorge

doi:10.1090/s0002-9939-2014-12077-6

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…Lacunary trigonometric functions appear in the context of the random Fourier series in harmonic analysis. Additionally, lacunary functions exhibit features upon approaching the natural boundary that are related to Weiner (stochastic) processes [22].…”

Section: Lacunary Sequencesmentioning

confidence: 99%

Cornu Spirals and the Triangular Lacunary Trigonometric System

Vogt

Ulness

2019

Fractal Fract

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This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed.

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Section: Lacunary Sequencesmentioning

confidence: 99%

Cornu Spirals and the Triangular Lacunary Trigonometric System

Vogt

Ulness

2019

Fractal Fract

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“…In molecular kinetic theory, lacunary functions display characteristics near the natural boundary that are related to Weiner (stochastic) processes. Because of this, lacunary functions have been studied in connection with Brownian motion (see Reference [10]).…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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.

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“…In kinetic theory, lacunary functions exhibit features upon approaching the natural boundary that are related to Weiner (stochastic) processes. As such, lacunary functions have been discussed in the context of Brownian motion [4,14,15].…”

Section: Introductionmentioning

confidence: 99%

Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

et al. 2019

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Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

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On the path properties of a lacunary power series

Cited by 6 publications

References 18 publications

Cornu Spirals and the Triangular Lacunary Trigonometric System

Cornu Spirals and the Triangular Lacunary Trigonometric System

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

Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

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