How to choose an orthonormal basis functions for approximating a specific problem

Question

Assume I have $f \in L^2(I)$, where $I=[0,a]$. After doing the weak formulation and Galerkin projection. I end up by approximating $f$ by $f^m$ such that $$ f\approx f^m=\sum_{i=1}^m f_i \Phi_i, $$ where $\{\Phi_i\}_{i=1}^m$ is an orthonormal basis function of $V_m$, a subspace of $L^2(I)$. There are many choices for the basis functions but is there a way or criterion to figure out the optimal choice? Thanks.

Answer 1

If I understand your question correctly, then I'd say that there is no "optimal" basis.

You are finding the point of the subspace that is closest to your given $f$. This closest point is independent of the basis used to express it. Different choices for the basis will give you different values for the coefficients $f_i$, but the approximant $f^m$ will be the same.

An analogy might help: suppose you have a plane $P$ in 3D space, and a point $Q$ that does not lie on $P$. You want to find the point of $P$ that's closest to $Q$. Obviously it does not matter what coordinate system you use to do the calculations. The closest point will have different coordinates in different coordinate systems, but it will still be the same point.