On H\"older continuity and $p^\mathrm{th}$-variation function of Weierstrass-type functions

Abstract

We study H\"older continuity, $p^\mathrm{th}$-variation function and Riesz variation of Weierstrass-type functions along the sequence of $b$-adic partitions, where $b>1$ is an integer. By a Weierstrass-type function, we mean that in the definition of the well-known Weierstrass function, the power function is replaced by a submultiplicative function, and the Lipschitz continuous cosine and sine functions are replaced by a general periodic H\"older continuous function.