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

TL;DR

This paper investigates the regularity and variation properties of generalized Weierstrass functions, focusing on their H"older continuity and p-th variation along b-adic partitions, extending classical results.

Contribution

It introduces a generalized class of Weierstrass functions with submultiplicative and H"older continuous components, analyzing their variation and continuity properties.

Findings

Established conditions for H"older continuity of generalized Weierstrass functions.

Analyzed p-th variation and Riesz variation along b-adic partitions.

Extended classical results to a broader class of fractal-like 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.