I'm looking to compute the Fourier Transform of the Weierstrass Function over the interval $[0,2\pi]$: $$W(t) = \sum_{n=1}^\infty a^n \cos(b^n t) $$ where $0<a<1$, $b$ is a positive integer such that and $ab > 1+\frac{3\pi}{2}$.
In computing the integral $$\int_0^{2\pi} \sum_{n=1}^{\infty} a^n\cos(b^nt)e^{2\pi i st}dt$$ I've tried writing $\cos(b^nt) = \frac{e^{ib^nt }+e^{-ib^nt}}{2}$. I use geometric series to get rid of the sum, but then I get the integral
$$\int_0^{2\pi} \frac{e^{2\pi i s t}}{2}e^{ibt} \left(\frac{1}{1/a-e^{ibt}} + \frac{1}{1/a-e^{-ibt}} \right)dt $$ which I don't know how to solve, even when I put it under a common demoninators and right the bracketed expression in terms of $\cos(bt)$.
I've also tried interchanging the sum and the integral. However, I get an unwiedly sum.
Is there a clever way of going about this or am I missing something in one of these methods.
For $b$ an integer $\ne -1,0,1$ and $|a|<1$ then $$W(t)=\sum_{n=1}^\infty a^n \cos(b^n t)$$ is already a $2\pi$-periodic Fourier series and $$ c_k=\frac1{2\pi}\int_0^{2\pi} W(t)e^{-ikt}dt = \begin{cases}\dfrac12 a^{n} & \text{if } |k| = |b|^n\\ \\ 0 &\text{otherwise}. \end{cases} $$ $$ W(t)=\sum_{k=-\infty}^\infty c_k e^{ikt}$$