Manfred Krüppel

The partial derivatives of de Rham's singular function and power sums of binary digital sums

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 66, 45 - 67 (2011)

MSC: 11A63   Radix representation; digital problems {For metric results, see 11K16}
  39B22   Equations for real functions {See also 26A51, 26B25}
  26A27   Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Abstract:   This note is a supplement to the paper \cite{kru-kr2} on the partial derivatives $T_n$ of de Rham's function $R_a(x)$ with respect to the parameter $a$ at $a=1/2$. In particular, $T_0(x)=x$ and $T_1(x)=2T(x)$ where $T$ is Takagi's continuous nowhere differentiable function. We present a new representation of $T_n$. From this we derive a limit relation at dyadic rational points. Moreover, we show that real linear combinations of $T_n$ with $n\ge1$ are nowhere differentiable. Thus we are able to prove that the functions which appear e.g. in the well known formula of Coquet for power sums of binary digital sums are nowhere differentiable. Finally, we derive a corresponding formula for power sums of the number of zeros.

Keywords:   De Rham's singular function, Takagi's continuous nowhere differentiable function, functional equations, binary digital sums, number of zeros, Stirling numbers.
Notes:   The author wishes to thank L. Berg for his hint to the Stirling numbers.
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