I am looking for a consistent theory (that is, something that is quotable, and that provides all relevant theorems) on the fact that a polynomial function can be reproduced by knowing its integration value on several independent domains. For instance it's clear that:
- A $K^{th}$ degree 1D polynomial function $f(x)$ can be exactly reproduced by knowing its integrals $\displaystyle I_i = \int_{S_i\in \Omega}{f(x)dx}$ over a set $\Omega$ of $K+1$ (consecutive) segments.
- A vector field of polynomial degree $K$ defined in a 2D domain can be reproduced by knowing its integrated divergence and curl over $K$ independent regions---In fact the integrated curl (resp. div) is enough to reproduce its coexact (resp. exact) components.
This sounds to me like some relatively basic property of finite-elements, or the method of moments, or just a variation on quadrature, but I couldn't find an exact reference uniting such properties.