Suppose the set $\mathscr{H} = \{\phi_i\}_{i=1}^{\infty}$ is a complete orthogonal set (of functions) with a given inner product operation. Let $\mathscr{H}_N = \{\phi_i\}_{i=1}^{N}$, i.e. the first $N$ elements composing the set $\mathscr{H}$.
I'm wondering under which conditions I can find a mapping $\Gamma:\mathscr{H}_N \rightarrow \mathscr{H}$ (of course nonlinear) such that I can generate every element of the set $\mathscr{H}$.
In other words, I would like to know under which conditions I can find a mapping such that, after its application to the finite number of (linearly independent) elements of $\mathscr{H}_N$, I can generate every $\{\phi_i\}_{i=1}^\infty$.
Thanks in advance!