About the Uniform Hölder Continuity of Generalized Riemann Function

Abstract

In this paper, we study the uniform Hölder continuity of the generalized Riemann function $R_{\hat{I}\pm ,{\hat{I}}^2}$ (with $\hat{I}\pm >1$ and ${\hat{I}}^2>0$) defined by \[ R_{α,β}(x)=\sum_{n=1}^{+\infty}\frac{\sin(πn^βx)}{n^α},\quad x\in\mathbb{R}, \] using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of $R_{\hat{I}\pm ,{\hat{I}}^2}$ as ${\hat{I}}^2$ tends to infinity.

17 pages, 2 figures