I'm struggling with solving the following problem.
Let $0<a<1$ and $2\leq b \in \mathbb{Z}$, so $b$ is an integer. Prove that the following series \begin{align} \sum_{k=0}^\infty a^k \cos(b^k x) \end{align} defines a function from $C^\alpha[-\pi,\pi]$ where $0<\alpha<1$ if $ab=1$.
First I proved the uniform convergence of this series. Using the absolute convergence and geometric series, we have \begin{align} \left\vert \sum_{k=0}^\infty a^k \cos(b^k x) \right\vert \leq \sum_{k=0}^\infty \vert a^k \cos(b^k x) \vert \leq \sum_{k=0}^\infty a^k = \frac{1}{1-a} < \infty, \end{align} hence the series is uniformly convergent. Therefore, we can express \begin{align} \sum_{k=0}^\infty a^k \cos(b^k x) - \sum_{k=0}^\infty a^k \cos(b^k y) = \sum_{k=0}^\infty a^k \left( \cos(b^k x) - \cos(b^k y) \right). \end{align} Assume that $x<y$ and $\vert x-y \vert < 1$, and then for each $k$, we have \begin{align} \vert \cos(b^k x) - \cos(b^k y) \vert \leq C \vert x-y \vert \vert \sin (b^k x_0) \vert \leq C \vert x-y \vert^\alpha, \end{align} for some $C>0$ and $x_0\in[x,y]$ by the mean value property. We don't have to consider the case $\vert x-y \vert \geq 1$ since $\vert x-y \vert$ is at most $2\pi$. So summing up all these so far, \begin{align} \left\vert \sum_{k=0}^\infty a^k \left( \cos(b^k x) - \cos(b^k y) \right) \right\vert \leq \frac{C}{1-a} \vert x-y \vert^\alpha. \end{align} Does this complete the proof?
Now I resolve the OP and post this answer. As mentioned in the OP, we just consider the case $\vert x-y \vert <1$. For a given $h=\vert x-y \vert$, there is $0<k_0\in\mathbb{Z}$ such that $a^{k_0+1} \leq \vert x-y \vert < 1/b^{k_0}=a^{k_0}$ by Archimedean property. Then we have \begin{align} \left\vert \sum_{k=0}^\infty a^k(\cos(b^k x) - \cos(b^k y)) \right\vert &\leq \left\vert \sum_{k\leq k_0} a^k(\cos(b^k x) - \cos(b^k y)) \right\vert + \left\vert \sum_{k>k_0} a^k(\cos(b^k x) - \cos(b^k y)) \right\vert \\ &\leq C\sum_{k\leq k_0} \vert x-y \vert + 2\sum_{k>k_0}a^k \\ &\leq C_1\vert x-y \vert^\alpha + C_2 a^{k_0+1} \leq Ch^\alpha, \end{align} using the mean value property for the first sum and the geometric series for the second sum. This completes the proof.