Piecewise Monotone Polynomial Approximation

Given a real function f satisfying a Lipschitz condition of order 1 on $[a,b]$, there exists a sequence of approximating polynomials \ Pₙ\ such that the sequence Eₙ = ||Pₙ - f|| (sup norm) has order of magnitude $1/n$ (D. Jackson). We investigate the possibility of selecting polynomials Pₙ having the same local monotonicity as f without affecting the order of magnitude of the error. In particular, we establish that if f has a finite number of maxima and minima on $[a,b]$ and S is a closed subset of $[a,b]$ not containing any of the extreme points of f, then there is a sequence of polynomials Pₙ such that Eₙ has order of magnitude $1/n$ and such that for n sufficiently large Pₙ and f have the same monotonicity at each point of S. The methods are classical.