An elementary simultaneous approximation theorem

Abstract

We will give an elementary and direct proof that for f ∈ C q [ − 1 , 1 ] f \in {C^q}[ - 1,1] there exists a sequence of polynomials P n {P_n} of degree at most n ( n > 2 q ) n\;(n > 2q) such that for k = 0 , … , q k = 0, \ldots ,q \[ | f ( k ) ( x ) − P n ( k ) ( x ) | ⩽ M q , k ( 1 − x 2 n ) q − k E n − q ( f ( q ) ) , |{f^{(k)}}(x) - P_n^{(k)}(x)| \leqslant {M_{q,k}}{\left ( {\frac {{\sqrt {1 - {x^2}} }} {n}} \right )^{q - k}}{E_{n - q}}({f^{(q)}}), \] with M q , k {M_{q,k}} depending only upon q q and k k . Moreover f ( q ) ( ± 1 ) = P n ( q ) ( ± 1 ) {f^{(q)}}( \pm 1) = P_n^{(q)}( \pm 1) .