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Oscillatory Property And Dimensions Of Rademacher Series

Author

Listed:
  • 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)

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.

Suggested Citation

  • Yuewei Pan & Shanfeng Yi, 2023. "Oscillatory Property And Dimensions Of Rademacher Series," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(05), pages 1-12.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:05:n:s0218348x23500378
    DOI: 10.1142/S0218348X23500378
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