Let $f(t) \in L^2$ and $\{\alpha_i(t)\}$ be a set of $N$ linearly independent functions in $L^2$. Let the $i$th entry of $b \in R^N$ equal $\int f(t) \alpha_i(t)dt$.
Is there a general method for solving for $f(t)$ given $b$? I realize that the solution is in general not unique, but I was wondering if there was a general way to find any such solutions (perhaps the minimum norm solution?).
Do a Gram-Schmidt orthogonalization of the $\alpha_i$. Let be $\{\beta_i\}_{i=1}^N$ the new subspace basis. Write the coefficients $c_i = \int f\beta_i$ depending on the $b_i = \int f\alpha_i$. The sum $\sum_{i=1}^N c_i\beta_i$ will be a solution of your problem.