A geometric construction of continuous, strictly increasing singular functions

Okamoto, Hisashi; Wunsch, Marcus

doi:10.3792/pjaa.83.114

Cited by 33 publications

(26 citation statements)

References 5 publications

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“…A by far not complete overview of the papers written about the Minkowski question mark function or closely related topics (Farey tree, enumeration of rationals, Stern's diatomic sequence, various 1-dimensional generalizations and generalizations to higher dimensions, statistics of denominators and Farey intervals, Hausdorff dimension and analytic properties) can be found in [1]. These works include [5], [6], [8], [9], [10], [12], [13] (this is the only paper where the moments of a certain singular distribution, a close relative of F (x), were considered), [11], [14], [16], [18], [20], [24], [25], [26], [27], [28], [29], [30], [31], [33]. The internet page [36] contains an up-to-date and exhaustive bibliographical list of papers related to the Minkowski question mark function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The Minkowski question mark function: explicit series for the dyadic period function and moments

Alkauskas

2010

Math. Comp.

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Abstract. Previously, several natural integral transforms of the Minkowski question mark function F (x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F (x). One of them, the dyadic period function G(z), was defined as a Stieltjes transform. In this paper we introduce a family of "distributions" F p (x) for p ≥ 1, such that F 1 (x) is the question mark function and F 2 (x) is a discrete distribution with support on x = 1. We prove that the generating function of moments of F p (x) satisfies the three-term functional equation. This has an independent interest, though our main concern is the information it provides about F (x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Direct inspection of the proof reveals that the statement of the lemma still holds with a weaker assumption that H(z) is a real-analytic function on (−∞, 0]. Now, let us differentiate (23) n times with respect to p, use (24) and afterwards substitute p = 2. This gives…”

Section: Three-term Functional Equationmentioning

confidence: 99%

The Minkowski question mark function: explicit series for the dyadic period function and moments

Alkauskas

2010

Math. Comp.

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“…( x x + 1 ), which is, of course, exactly F(x). Okamoto and Wunsch [31] construct yet another generalisation of ? (x), though their main concern is to introduce a new family of purely singular functions.…”

Section: Giedrius Alkauskasmentioning

confidence: 99%

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

Alkauskas

2009

Glasgow Math. J.

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Abstract. The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane ‫ރ‬ \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert-Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern-Brocot tree. Surprisingly, the Eisenstein series G 2 (z) does manifest in both real and p-adic cases.

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“…H. Okamoto had also generalized Salem's method in his paper [3] and [4] to obtain more singular functions and continuous nowhere differntiable functions.…”

Section: F Is Continuousmentioning

confidence: 99%

“…It seems to be very strange that a continuous increasing function is singular. But there are even strictly increasing continuous singular functions (see, for example, [4] and [5]). It's well known that all the derivatives of the boundary functions of strictly convex divisible (or quasihomogeneous) projective domains are such functions if the domain is not an ellipse (see [1]).…”

Section: Introductionmentioning

confidence: 99%