L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial pm,n∗ of degree m which interpolates a given real-valued function f ∈ L2[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L2-sense over all polynomials of degree ⩽ m with the same interpolatory character. It is shown that the L2-error ‖pm,n∗ − f‖ → 0 as m → ∞ if f ∈ C[a, b].