SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

Title & Authors
SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION
Baek, In-Soo;

Abstract
We give the characterization of H$\small{\ddot{o}}$lder differentiability points and non-differentiability points of the Riesz-N$\small{\acute{a}}$gy-Tak$\small{\acute{a}}$cs (RNT) singular function $\small{{\Psi}_{a,p}}$ satisfying $\small{{\Psi}_{a,p}(a)=p}$. It generalizes recent multifractal and metric number theoretical results associated with the RNT function. Besides, we classify the singular functions using the singularity order deduced from the H$\small{\ddot{o}}$lder derivative giving the information that a strictly increasing smooth function having a positive derivative Lebesgue almost everywhere has the singularity order 1 and the RNT function $\small{{\Psi}_{a,p}}$ has the singularity order $\small{g(a,p)=\frac{a{\log}p+(1-a){\log}(1-p)}{a{\log}a+(1-a){\log}(1-a)}{\geq}1}$.
Keywords
Hausdorff dimension;packing dimension;distribution set;local dimension set;singular function;metric number theory;H$\small{\ddot{o}}$lder derivative;
Language
English
Cited by
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