Given the sequence $\{ f(i) \}_{0 \leq i \leq n}$ and the array $\{ g_{n}(i,j) \}_{0 \leq i, j \leq n}$ such that $f(i) = \sum_{j = 0}^{n} h_{j} \, g_{n}(i,j)$ for some integers $h_{0}, \dots, h_{n}$, is there a way to write $h_{j}$ as a sum over $f(i)$ and $g_{n}(i,j)$ using the calculus of finite differences?
Thanks.
Update: The first post below answers the question of writing $h_{j}$ as a function of $f$ and $g$, but I'd like to avoid matrix inversion if at all possible, hence calculus of finite differences.