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Fractal Fract 2018, 2(1), 2; https://doi.org/10.3390/fractalfract2010002

Fractal Curves from Prime Trigonometric Series

1
NIKI Ltd. Digital Engineering, Research Center, 205 Ethnikis Antistasis Street, 45500 Katsika, Ioannina, Greece
2
TWT GmbH Science & Innovation, Mathematical Research, Ernsthaldenstr. 17, 70565 Stuttgart, Germany
*
Author to whom correspondence should be addressed.
Received: 10 November 2017 / Revised: 26 December 2017 / Accepted: 26 December 2017 / Published: 3 January 2018
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Abstract

We study the convergence of the parameter family of series: V α , β ( t ) = p p α exp ( 2 π i p β t ) , α , β R > 0 , t [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically. View Full-Text
Keywords: trigonometric series; lacunary series; Hölder continuity; fractality; random Fourier series; prime distribution trigonometric series; lacunary series; Hölder continuity; fractality; random Fourier series; prime distribution
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Vartziotis, D.; Bohnet, D. Fractal Curves from Prime Trigonometric Series. Fractal Fract 2018, 2, 2.

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