Consider the sum $$ \sum_{k=1}^n a_{k} \exp\left(\frac{ick}{n}\right). $$
I have heard of methods that treat exponential sums. I was wondering if it's possible to find general expressions for exponential sums with "weights". Note that $a_{k}$ in general is a decreasing function of $k$.
Note that c is any constant and assume that $n\to\infty$ .
Your $a_k$ are strictly related with the Fourier series, expressed in the complex form.
Since you put $n$ at the denominator of the argument of $\exp $, when $n \to \infty$ you are moving towards a Rieman sum and thus towards the Fourier integral transform, if you can "extract" the $1 / n$ coefficient from the $a_k$'s, while maintaining the required convergence criteria for the Fourier series/transform.
