In Shamir's secret sharing and Reed Solomon codes we see that polynomials are good candidates for hiding data or correcting errors, the former is taught in graduate courses. This made me ponder about special cases such as LRC codes or the gaussian quadrature, allowing the use of less data for accurate reconstruction (or quadrature evaluation). Most cases and methods I know and tried, minimize the error and not deal with perfect reconstruction. Having said all that, here is my problem statement:
I want to find $\int_{-1}^{1}f(x)w(x)dx$ numerically with $f(x)$ as a polynomial of degree $2m-1$ and $w(x)$ is a weight function. For a polynomial of degree $2m-1$ we can use only $m$ specific evaluation points to find the quadrature precisely. We can also use any $2m-1$ distinct points to find the result. Is there a middle ground? given $f(x)$ as defined above, can I find $k$ "special" points such that $m<n<k<2m-1$ (strictly smaller) every $n$ out of $k$ would return the result accurately?