This is a revision question I've been working on.
Show that if a $2\pi$-periodic function $f$ has the complex Fourier coefficients $c_{k}$ and $g(t)=f(t+a)$, where $a$ is a constant, the the Fourier coefficients $y_{k}$ of $g$ and given by $y_{k}=e^{ika}c_{k}$.
Now suppose $f$ has Fourier coefficients $c_{k}=e^{-k^{2}}$ and $g$ has Fourier coefficients $p_{k}=(1+k^{2})^{-1}$. Define $h(t)=2f(t+1)-g(t-2)-3$. Find the complex Fourier coefficients of $h$.