Let $\gamma_1\left(x\right) = \sum_{n=1}^{N} \alpha_n f_n\left(x\right)$ where $f_n\left(x\right)$ are orthonormal under an inner product over a range $\left[x_1, x_2\right]$. I already know all of the $f_n$ and can compute $\gamma\left(x\right)$ at all $x$ in the range.
Now I would like to compute $\gamma_2\left(x\right) = \sum_{m=1}^{M} \beta_m g_m\left(x\right)$ where $g_m\left(x\right)$ are orthogonal under the same inner product over the range $\left[x_1, x_2\right]$. The difference is that I must enforce $\gamma_2\left(x\right) = 0$ at pre-specified $x$ values.
Is there a way to determine this orthogonal set of functions $g_m$ given that I know $f_n$ while also satisfying the restrictions on $\gamma_2\left(x\right)$?
My application is actually related to matrices, their eigenvectors, and what happens to eigenvectors of reduced forms of the original matrix, but I'm curious if the question can be more generally answered. Hopefully, I've described it sufficiently.