Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

  • Baek, In-Soo (Department of Mathematics Busan University of Foreign Studies)
  • Received : 2014.07.29
  • Published : 2015.01.31
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Abstract

We give the characterization of H$\ddot{o}$lder differentiability points and non-differentiability points of the Riesz-N$\acute{a}$gy-Tak$\acute{a}$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$\ddot{o}$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 ${\Psi}_{a,p}$ 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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References

  1. I. S. Baek, Relation between spectral classes of a self-similar Cantor set, J. Math. Anal. Appl. 292 (2004), no. 1, 294-302. https://doi.org/10.1016/j.jmaa.2003.12.001
  2. I. S. Baek, Dimensions of distribution sets in the unit interval, Commun. Korean Math. Soc. 22 (2007), no. 4, 547-552. https://doi.org/10.4134/CKMS.2007.22.4.547
  3. I. S. Baek, Multifractal spectrum in a self-similar attractor in the unit interval, Commun. Korean Math. Soc. 23 (2008), no. 4, 549-554. https://doi.org/10.4134/CKMS.2008.23.4.549
  4. I. S. Baek, Derivative of the Riesz-Nagy-Takacs function, Bull. Korean Math. Soc. 48 (2011), no. 2, 261-275. https://doi.org/10.4134/BKMS.2011.48.2.261
  5. I. S. Baek, L. Olsen, and N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007), no. 1, 267-287. https://doi.org/10.1016/j.aim.2007.02.003
  6. P. Billingsley, Probability and Measure, John Wiley and Sons, 1995.
  7. C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on ${\mathbb{R}}$, Intern. J. Math. Math. Sci. 11 (1988), no. 4, 643-649. https://doi.org/10.1155/S016117128800078X
  8. R. Darst, The Hausdorff dimension of the non-differentiability set of the Cantor function is $(\frac{ln\;2}{ln\;3})^2$, Proc. Amer. Math. Soc. 119 (1993), no. 1, 105-108. https://doi.org/10.1090/S0002-9939-1993-1143222-3
  9. R. Darst, Hausdorff dimension of sets of non-differentiability points of Cantor functions, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 1, 185-191. https://doi.org/10.1017/S0305004100073011
  10. J. Eidswick, A characterization of the non-differentiability set of the Cantor function, Proc. Amer. Math. Soc. 42 (1974), 214-217. https://doi.org/10.1090/S0002-9939-1974-0327992-8
  11. K. J. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, 1997.
  12. M. Kessebohmer and B. O. Stratmann, Holder-differentiability of Gibbs distribution functions, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 489-503. https://doi.org/10.1017/S0305004109002473
  13. H. H. Lee and I. S. Baek, A note on equivalent interval covering systems for packing dimension of ${\mathbb{R}}$, J. Korean Math. Soc. 28 (1991), no. 2, 195-205.
  14. L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 43-53. https://doi.org/10.1017/S0305004104007601
  15. L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc. (2) 67 (2003), no. 1, 103-122. https://doi.org/10.1112/S0024610702003630
  16. J. Paradis, P. Viader, and L. Bibiloni, The derivative of Minkowski's ?(x) function, J. Math. Anal. Appl. 253 (2001), no. 1, 107-125. https://doi.org/10.1006/jmaa.2000.7064
  17. J. Paradis, P. Viader, and L. Bibiloni, Riesz-Nagy singular functions revisited, J. Math. Anal. Appl. 329 (2007), no. 1, 592-602. https://doi.org/10.1016/j.jmaa.2006.06.082
  18. H. L. Royden, Real Analysis, Macmillan Publishing Company, 1988.