Polynomial approximation using projections whose kernels contain the Chebyshev polynomials

Let P be any projection of f ε C^{n + 1}[−1, 1] onto _n such that P(T_{n + 1}) = 0. It is shown that, for a wide class of such projections, f − Pf may be expressed in terms of f^{n + 1} in the same manner as E_n(f), the error of minimax approximation. Furthermore, a general necessary condition is obtained for this phenomenon.