I'm struggling into show the relation between $\hat g[k]$ and $\hat h[k]$ given $g[n]=(-1)^nh[n]$, where $h[n]$ is a discrete signal, let's say with $0\le n <N$. I made some computations that I show you
\begin{align*} \hat g[k]&=\sum\limits_{n=0}^{N-1}g[n]e^{-i2\pi kn/N}=\sum\limits_{n=0}^{N-1}(-1)^nh[n]e^{-i2\pi kn/N} \\ &=\sum\limits_{n \text{ even}}h[n]e^{-i2\pi kn/N}-\sum\limits_{n \text{ odd}}h[n]e^{-i2\pi kn/N} \end{align*}
and the same hold also for $\hat h[k]$ since the functions are switchable
\begin{align*} \hat h[k]&=\sum\limits_{n=0}^{N-1}h[n]e^{-i2\pi kn/N}=\sum\limits_{n=0}^{N-1}(-1)^ng[n]e^{-i2\pi kn/N} \\ &=\sum\limits_{n \text{ even}}g[n]e^{-i2\pi kn/N}-\sum\limits_{n \text{ odd}}g[n]e^{-i2\pi kn/N} \end{align*}
Now from here I tried various ways but I cannot find a fine relation between them. I ended up with these
$$ \hat g[k]+\hat h[k] = 2\sum\limits_{n \text{ even}}h[n]e^{-i2\pi kn/N} $$
$$ \hat g[k]-\hat h[k] = -2\sum\limits_{n \text{ odd}}h[n]e^{-i2\pi kn/N} $$
But I'm not sure if these can be considered as a solution, maybe I miss something else. Thanks in advance.