Simultaneous best approximation by three polynomials

Abstract

Let [a, b] be any interval and let p₀, p₁, p_k be any three polynomials of degrees 0, 1, k, respectively, where k ≥ 2. A set of necessary and sufficient conditions for the existence of an f in C[a, b] such that p_i is the best approximation to f from the space of all polynomials of degree less than or equal to i, for all i = 0, 1, k, is given.