Nearly Coconvex Approximation

* Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001

Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are nearly coconvex with it, namely, polynomials and piecewise polynomials that preserve the convexity of f except perhaps in some small neighborhoods of the points where f changes its convexity. We obtain Jackson type estimates and summarize the positive and negative results in a truth-table as we have previously done for nearly comonotone approximation.