Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions
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Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions E. de Amo · M. Díaz Carrillo · J. Fernández-Sánchez
Received: 16 December 2010 / Accepted: 6 December 2011 / Published online: 17 December 2011 © Springer Science+Business Media B.V. 2011
Abstract We study a class of singular functions via a generalized dyadic system and Hausdorff dimensions are calculated for several sets related with these functions. Furthermore, we introduce a class of monotonic type on no-interval and almost everywhere differentiable functions that includes—as an exceptional case—the continuous nowhere differentiable Takagi function (multiplied by 2) among them. Keywords Singular function · Generalized dyadic system · Schauder basis · (Simply) normal number · Hausdorff (fractal) dimension · MTNI function
1 Introduction The first example of a continuous nowhere differentiable function was published by du BoisReymond in 1875. This example, given by the formula Wa,b (x) :=
+∞
a k cos bk πx ,
k=0
3 0 < a < 1, ab > 1 + π, b + 1 ∈ 2Z, 2
was due to Weierstrass. Afterwards, Hardy proved that it is still a continuous nowhere differentiable function if 0 < a < 1, ab > 1. E. de Amo () · J. Fernández-Sánchez Universidad de Almería, Almería, Spain e-mail: [email protected] J. Fernández-Sánchez e-mail: [email protected] M. Díaz Carrillo Universidad of Granada, Granada, Spain e-mail: [email protected]
130
E. de Amo et al.
Later, in 1903, Takagi gave another example of a continuous nowhere differentiable function as follows: T (x) :=
+∞ d(2k x) k=0
2k
,
∀x ∈ R,
(1)
where d(x) denotes the distance of x from the nearest integer. Several properties of this function have been studied in depth, for example one-side derivatives, maxima, level sets, etc. as can be seen in [5, 6, 13, 24]. The first example of a singular function (i.e. a monotone increasing and continuous function whose derivatives vanish a.e.) was independently published by Cantor and Scheefer twelve years after the functions Wa,b were introduced. In 1904, Minkowski gave an example that allowed him to enumerate the quadratic irrational numbers. Moreover, he established a bijection between rationals and numbers in the unit interval I := [0, 1] whose dyadic representation is finite via Farey’s sequence. Denjoy showed the relation between Minkowski’s representation system for real numbers and the representation by simple continuous fractions. More recently, Viader et al. in [30], showed this function as the asymptotic distribution function of an enumeration of the rationals in I. The family of functions {Sa } we are going to study were introduced, simultaneously, by Césaro in 1906 and Hellinger in 1907. They have been studied from a wide variety of viewpoints (for example, geometric, arithmetic, probabilistic, or as functional equations), as can be seen in [3, 7, 10, 20, 27–29, 31]. One application for plastic deformation can be found in [8]. Other related references with respect to these functions can be found in [21]. In 1984, Hata and Yamagu
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