Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly independent, but not necessarily orthogonal). How does one systematically determine the set of functions $\{\psi_j(x)\}_{j=0}^N$ such that $$a_n = \sum_{j=0}^Np_j\psi_n(x_j).$$
As an example, in the Fourier transform $\phi_n(x_j) = e^{inx_j}$, where $x_j = 2\pi j/N$, and the inverting set is simply $\psi_j(x)\ = \frac{1}{N} e^{-ijx_n}$.
I am unfamiliar with this field, so please forgive any faux pas. As an additional question, what are these types of problems called, and is there any suggested reading on them?