关于缺项级数的变差函数
On variation functions of lacunary series
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中文总结 AI 辅助
本文研究沿$b$-adic划分的缺项级数变差函数，通过鞅中心极限定理推广了已有结果，适用于Takagi函数等周期函数。
中文摘要 AI 辅助
我们研究了沿$b$-adic划分序列的某些缺项级数的变差函数，其中$b>1$为整数。这里的缺项级数指的是，在众所周知的Weierstrass函数的定义中，将Lipschitz连续的余弦和正弦函数替换为一个连续且分段连续可微的周期函数，该函数的Fourier级数对所有非零整数$j$具有零$(jb)^{\mathrm{th}}$阶Fourier系数。在证明中，我们采用概率方法，应用鞅中心极限定理。我们的结果推广了Han、Schied和Zhang（2021）的一些近期结果，并且当所考虑的周期函数例如选择为Takagi函数、Weierstrass函数或某个缺项级数本身时，该结果均可适用。
英文摘要
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 $(jb)^{\mathrm{th}}$-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.
发表机构
- HUN-REN–SZTE Analysis and Applications Research Group, Bolyai Institute, University of Szeged（塞格德大学博莱伊研究所）
- Faculty of Mathematics and Natural Sciences, Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学数学与自然科学学院）
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