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We give the characterization of H\"older differentiability points and non-differentiability points of the Riesz-N{\'a}gy-Tak{\'a}cs (RNT) singular function $\Psi_{a,p}$ satisfying $\Psi_{a,p}(a)=p$. 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\"older 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 $\Psi_{a,p}$ has the singularity order $g(a,p)= \frac{a\log p +(1-a)\log (1-p)}{a\log a+(1-a) \log (1-a)}\geq 1$.
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참고문헌 (18)
참고문헌 신청
C. D. Cutler / 1988 / A note on equivalent interval covering systems for Hausdorff dimension on R, Intern / J. Math. Math. Sci 11 (4) : 643 ~ 649
C. D. Cutler / 1995 / Hausdorff dimension of sets of non-differentiability points of Cantor functions / Math. Proc. Cambridge Philos. Soc 117 (1) : 185 ~ 191
H. H. Lee / 1991 / A note on equivalent interval covering systems for packing dimension of R / J. Korean Math. Soc 28 (2) : 195 ~ 205
H. L. Royden / 1988 / Real Analysis / Macmillan Publishing Company
I. S. Baek / 2004 / Relation between spectral classes of a self-similar Cantor set / J. Math. Anal. Appl 292 (1) : 294 ~ 302
C. D. Cutler / 1995 / Hausdorff dimension of sets of non-differentiability points of Cantor functions / Math. Proc. Cambridge Philos. Soc 117 (1) : 185 ~ 191
H. H. Lee / 1991 / A note on equivalent interval covering systems for packing dimension of R / J. Korean Math. Soc 28 (2) : 195 ~ 205
H. L. Royden / 1988 / Real Analysis / Macmillan Publishing Company
I. S. Baek / 2004 / Relation between spectral classes of a self-similar Cantor set / J. Math. Anal. Appl 292 (1) : 294 ~ 302
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