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
Borel’s 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.

karin.martin@uni-rostock.de

Seite generiert am 06.09.2013, 11:54 Uhr