×

A Salem generalised function. (English) Zbl 1399.26008

Summary: Among the members of the celebrated family of functions introduced by Salem in the mid 20th century, there is a particular and very interesting one that we use to relate the dyadic system of numbers representation with the modified Engel system. Various properties are studied for this function, including derivatives and fractal dimensions.

MSC:

26A30 Singular functions, Cantor functions, functions with other special properties
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cantor G.: De la puissance des ensembles parfait de points. Acta Math. 4, 381–392 (1884) · JFM 16.0460.01 · doi:10.1007/BF02418423
[2] H. Lebesgue, Leçons sur l’integration et la Recherche des Fonctions Primitives, Gauthiers-Villars (Paris, 1904).
[3] P. Billingsley, Probability and Measure (2nd ed.), Wiley (New York, 1995). · Zbl 0822.60002
[4] H. Minkowski, Verhandlungen des III Internationalen Mathematiker-kongresses in Heidelberg (1904).
[5] Viader P., Paradís J., Bibiloni L.: A new light on Minkowski’s ?(x) function. J. Number Theory 73, 212–227 (1998) · Zbl 0928.11006 · doi:10.1006/jnth.1998.2294
[6] Paradís J., Viader P., Bibiloni L.: On actually computable bijections between \({\mathbb{N}}\) N and \({\mathbb{Q}^{+}}\) Q + . Order 13, 369–377 (1996) · Zbl 0941.11004 · doi:10.1007/BF00405596
[7] Paradís J., Viader P., Bibiloni L.: A total order in (0,1] defined through a ’Next’ Operator. Order 16, 207–220 (1999) · Zbl 0981.11030 · doi:10.1023/A:1006441703404
[8] E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, Harmonic analysis on the Sierpinski gasket and singular functions, Acta Math. Hungar., 143 (2013), 58–74. · Zbl 1324.28011
[9] Bohnstengel J., Kesseböhmer M.: Wavelets for iterated function systems. J. Funct. Anal. 259, 583–601 (2010) · Zbl 1196.42029 · doi:10.1016/j.jfa.2010.04.014
[10] E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, Singular functions with applications to fractals and generalised Takagi’s functions, Acta Appl. Math., 119 (2012), 129–148.
[11] E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, On duality of aggregation operators and k-negations, Fuzzy Sets and Systems, 181 (2011), 14–27. · Zbl 1239.39016
[12] Salem R.: On some singular monotone functions which are strictly increasing. Trans. Amer. Math. Soc. 53, 427–439 (1943) · Zbl 0060.13709 · doi:10.1090/S0002-9947-1943-0007929-6
[13] Paradís J., Viader P., Bibiloni L.: Riesz–Nagy singular functions revisited. J. Math. Anal. Appl. 329, 592–602 (2007) · Zbl 1115.26001 · doi:10.1016/j.jmaa.2006.06.082
[14] Takács L.: An increasing continuous singular function. Amer. Math. Soc. 85, 35–36 (1978) · Zbl 0394.26005 · doi:10.2307/2978047
[15] Rényi A.: A new approach to the theory of Engel’s series. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 5, 25–32 (1962)
[16] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (4th ed.), England Clarendon Press (Oxford, 1979). · Zbl 0423.10001
[17] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. (New York, 1970). · Zbl 0954.26001
[18] K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley&Sons (Chichester, 1990).
[19] Berg L., Kruppel M.: De Rham’s singular function and related functions. Z. Anal. Anwend. 19, 227–237 (2000) · Zbl 0985.39020 · doi:10.4171/ZAA/947
[20] E. Hewitt and K. R. Stromberg, Real and Abstract Analysis, Springer-Verlag (New York, 1965). · Zbl 0137.03202
[21] Kairies H. H.: Functional equations for peculiar functions. Aequationes Math. 53, 207–241 (1997) · Zbl 0876.39004 · doi:10.1007/BF02215973
[22] Wen L.: An approach to construct the singular monotone functions by using Markov chains. Taiwanese J. Math. 2, 361–368 (1998) · Zbl 0930.26004 · doi:10.11650/twjm/1500406976
[23] E. de Amo and J. FernándezSánchez, A generalised dyadic representation system, Int. J. Pure Appl. Math., 52 (2009), 49–66. · Zbl 1179.26092
[24] G. de Rham, Sur quelques courbes definies par des aequations fonctionnelles, Univ. Politec. Torino. Rend. Sem. Mat., 16 (1956), 101–113.
[25] Erdos P.: On the smoothness of the asymptotic distribution of additive arithmetical functions. Amer. J. Math. 61, 722–725 (1939) · Zbl 0022.01001 · doi:10.2307/2371327
[26] H. Niederreiter and L. Kuipers, Uniform Distribution of Sequences, John Wiley&Sons (New York, 1974). · Zbl 0281.10001
[27] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer (Berlin, 1976). · Zbl 0322.10002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.