Let $H=L^2(0,1)$ and $M$ be a linear manifold generated by $n$ independent functions, $\{b_1, \cdots, b_n\}$. Define the orthogonal projection $P_M : H \rightarrow M$ on $H$ into $M$. Then, we have for any $g \in H$, $$ P_M g = \sum_{i=1}^n \langle g, b_i^*\rangle b_i $$ where $b_i^*$ is the dual basis for $M$ of $b_i$. Generally, the computation of $b_i^*$ is very difficult because it requires the calculation of all pairs $\langle b_i, b_j \rangle$ unless $b_i$ is very simple or an orthonormal basis.
This difficulty motivates me to come up with some approximation method. Consider an orthonormal basis for $H$, say $\{e_i\}$. Then, $\{b_i\}$'s are expressed by an infinite series in terms of $\{e_i\}$, and define the truncated version of the infinite series, $\{\widetilde b_i\}$ defined by $$ \widetilde b_i = \sum_{k=1}^N \alpha_{ik}e_k. $$ Assume that we are able to compute $\alpha_{ij}$. Then, we can easily compute $\langle \widetilde b_i, \widetilde b_j \rangle $, allowing us to compute the dual basis $\widetilde b_i^*$ of $\widetilde b_i$. Consequently, our hope is that $\widetilde b_i^* \approx b_i^*$ as $N$ increases.
I think this approach is surely legitimate. So, I am highly sure that it is well studied and established about the error, convergence and so on. But I cannot find any proper keywords... Anybody knows this kind of approaches??