I am solving the following linear system:
$$c_q=\sum_ka_{k,q} f_k\\ a_{k,q}=\exp\left({2 \pi i\frac{k}{q+1}}\right)$$
with $ 0\leq k\leq m,0\leq q\leq m$. For this, it would be useful to calculate its determinant, eigenvalues, eigenvectors and inverse. It seems to me that such a matrix should be well documented in the literature, however I could not find a reference.
What is the solution to the above linear system?
So far I have the following: Numerical evidence suggests that $\lim_{m\to\infty}\det(a_{k,q})=0$ and $\forall m, \det(a_{k,q})\neq0$. Therefore, it seems that the homogeneous problem, $c_q=\vec{0}$, has only the trivial solution $f_k=\vec{0}$ unless we are dealing with the infinite case.
EDIT
Thanks to the comment by Paul Garret, I have found that for this Vandermonde matrix, we have
$$\det(a)=\prod_{1\le j<l\le n} (\exp\left(2 \pi i/l \right)-\exp\left(2 \pi i/j \right))$$
Similarily, this article gives the matrix inverse. This allows to find the seeked solution at least for the finite dimensional case, since the determinant is never null. However, does the solution hold when $n\to\infty$?.