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

Fractal Curves from Prime Trigonometric Series

NIKI Ltd. Digital Engineering, Research Center, 205 Ethnikis Antistasis Street, 45500 Katsika, Ioannina, Greece
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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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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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Vartziotis, D.; Bohnet, D. Fractal Curves from Prime Trigonometric Series. Fractal Fract 2018, 2, 2.

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