Polynomial approximation using projections whose kernels contain the Chebyshev polynomials

Abstract

Let P be any projection of f ϵ C n + 1 [−1, 1] onto P 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.