OSCILLATORY PROPERTY AND DIMENSIONS OF RADEMACHER SERIES

Yuewei Pan () and Shanfeng Yi
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Yuewei Pan: School of Mathematics Sciences, GuiZhou Normal University, Guizhou, Guiyang 550001, P. R. China
Shanfeng Yi: School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, P. R. China

FRACTALS (fractals), 2023, vol. 31, issue 05, 1-12

Abstract: Let ∑i=1∞c iRi(x) be the Rademacher series, where {Ri(x)}i=1∞ is the classical Rademacher function system and {ci}i=1∞ is an arbitrary real number sequence. In this paper, we first show that the value range of the Rademacher series at any subinterval of [0, 1] is ℠∪{±∞} when {ci}1∞∈ ℓ2∖ℓ1. This result provides us with the basic facts that when {ci}1∞∈ ℓ2∖ℓ1, the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of [0, 1]. Further, when {ci}1∞∈ ℓ2∖ℓ1, we show various dimensions of the level set of Rademacher series on any subinterval of [0, 1]. Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when {ci}1∞∈ ℓ1, and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.

Keywords: Radermacher Series; Oscillatory Property; Local Level Set; Box Dimension; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23500378

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