I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress.
I need to show that the function $f(t) = \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\in\Lambda^{\alpha}$ when $0 < \alpha \leq 1$. The second part is to show that if $\alpha < \beta < 1$, then $f\notin\Lambda^{\beta}$, but I'll worry about the second part later.
I tried considering $ \begin{eqnarray*} |f(t+h) - f(t)| &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}(t + h))}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t + 3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h) - \cos(3^{n}t)}{3^{n\alpha}}\right|\\ &\leq& \sum_{n=1}^{\infty}\left|\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h) - \cos(3^{n}t)}{3^{n\alpha}}\right|\\ \end{eqnarray*} $
EDIT: Removed the last half - dozen lines which turned out to be completely non-constructive.
Now I'm not sure if I'm even remotely close to going down the right path, but if I could get this manipulated into something of the form $C^{\alpha}$ I'd be done. But I just can't seem to go any further. Any suggestions?