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On variation functions of lacunary series

  • Matyas Barczy
  • Peter Kern
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Abstract

We study variation functions of some lacunary series along the sequence of b-adic partitions, where b>1 is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero (jb)th-Fourier coefficients for all non-zero integers j. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.

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Cite

BibTeXRIS
BibTeX
@misc{barczy2026variation,
  title = {{On variation functions of lacunary series}},
  author = {Matyas Barczy and Peter Kern},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22498},
  eprint = {2609.22498},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22498},
}
RIS
TY  - UNPB
TI  - On variation functions of lacunary series
AU  - Matyas Barczy
AU  - Peter Kern
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.22498
UR  - https://arxiv.org/abs/2609.22498
AB  - We study variation functions of some lacunary series along the sequence of b-adic partitions, where b>1 is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero (jb)^th-Fourier coefficients for all non-zero integers j. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.
ER  -