I'm having trouble seeing if the following series converges:
Let $g$ be an analytic $1$-periodic function. Let $g(x) = \sum_k g_k e^{ikx}$ be its Fourier expansion. Recall that the Fourier coefficients of an analytic function have exponential decay, that is $g_k \leq M e^{-\delta k}$ for some $M$ and $\delta$.
We define $h = \sum_k h_k e^{ikx}$,
Where $h_k = \frac{-g_k}{2}$ if $k$ is odd. If $k$ is even, $k = 2n$ then $h_{2n} = \frac{h_n - g_{2n}}{2}$.
I see that for $k$'s that are finitely divisible by $2$, $h_k$ is a finite sum of Fourier coefficients so we should have no problem since those decrease exponentially. Consider the case where $k$ is a perfect power of $2$ $k = 2^n$
Then we have
\begin{equation} h_{2^k} = \frac{1}{4}\sum_{i=1}^k \frac{g_i}{2^{k-i}} \end{equation}
It is clear that $h_2^k \to 0$. How can I see if $ lim_K \sum_k^K h_{2^k}$ converges or diverges?
I have the feeling that it diverges since we have a sum of something that behaves like
\begin{equation} \frac{1}{2^k}(\sum_i^k (2/e^\delta)^i ) \end{equation}
but I'm failing to produce a proof