I have two ON bases where I to the best $L_2$ precision possible represent the same function. Let us say that these have $n_1,n_2$ basis functions respectively for which we know the coefficient perfectly.
Example: I have a function which has both a Discrete Fourier Transform and a power series expansion around on $[-1,1]$ and around $0$ respectively. Say I have both all Fourier coefficients up to $n_1$ and all Monomial coefficients up to $n_2$.
How do I find which family of functions which linear combinations best (in some sense) preserve this knowledge, up to, say $n_3$ terms? I will assume we want $n_3\lt n_2+n_1$, since it would otherwise be quite trivial, as we then could simply stack the representations on top of each other as we don't need to "compress" anything.