Prove Holder-continuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$

Consider $\lambda \in (0,1]$ and the function $\mu_\lambda$ defined by $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$ for $x \in [0,\pi]$.  Show that  $\mu_\lambda$  is  $\alpha$-Holder-continuous on $[0, \pi]$ for all  $0 < \alpha < \lambda$.

  • This is a tricky question. The offered bounty is low.

  • I would double the bounty.


Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
  • This took me over an hour to figure out the solution and and write it down. Please offer higher bounties in future, otherwise you may not get a response.

The answer is accepted.