Manfred Krüppel

On the Solutions of Two-Scale Difference Equations

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 67, 59 - 88 (2012)

MSC: 39A13   Difference equations, scaling ($q$-differences) [See also 33Dxx]
  26A16   Lipschitz (Hölder) classes
  26A24   Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
  26A27   Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
  11K16   Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]

Abstract:   This paper deals with specific two-scale difference equations which are equivalent to a system of functional equations. Such equations have a continuous solution if the coefficients cj of the corresponding characteristic polynomial P satisfy condition |cj | < 1 for all j. By means of some functional relations for the solution we show that it is Hölder continuous and we determine the optimal Hölder exponent. Moreover we give a condition which is necessary and sufficient for the differentiability almost everywhere where we apply Borelís normal number theorem. If the coefficients cj are nonnegative then the solution is a singular function. Special cases are the well-known singular functions of de Rham and of Cantor.

Keywords:   Difference equations, scaling
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