On approximation of a function and its derivatives by a polynomial and its derivatives

Let g∈C ^q ?1,1] be such thatg ^(k)(±1)=0 fork=0,…,q. LetP _n be an algebraic polynomial of degree at mostn, such thatP _n ^(k) (±1)=0 for $k = 0, \cdots ,\left {\begin{array}{*{20}c} {q + 1} \\ 2 \\ \end{array} } \right]$ . The,P _n and its derivativesP _n ^(k) fork≤q well approximateg and its respective derivatives, provided only thatP _n well approximatesg itself in the weighted norm $$\left\| {\frac{{g(x) - P_n (x)}}{{(\sqrt {1 - x^2 } )^4 }}} \right\|.$$ This result is easily extended to an arbitraryf∈C ^q ?1,1], by subtracting fromf the polynomial of minimal degree which interpolatesf ⁽⁰⁾,…,f ^(q) at±1. As well as providing easy criteria for judging the simutaneous approximation properties of a given polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation inC ^q ?1,1] and trigonometric polynomial approximation in the space ofq times differentiable 2x-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation procedures for small values ofq.