Given a table with x and f(x) values, g(x) a non-linear least squares approximation, and p(x) the Lagrange interpolating polynomial that passes through all of the points in the given table. What would be the best approximation on f, for a point x = k by only using p(x) and g(x)?
Intuitively I thought about calculating p(k) and g(k), then taking the middle point of both, however I have no idea if this is reasonable to think of, nor if it is the best approximation for f(k). Is there a visual way to think of this problem? Or will I have to use the properties of both p and g to find the optimal approximation?