Geodesic
Nearly Coconvex Approximation
Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 361-378.

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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.
Keywords: Nearly Coconvex Approximation, Polynomial Approximation, Piecewise Polynomial Approximation, Jackson Estimates 
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Leviatan, D.; Shevchuk, I. Nearly Coconvex Approximation. Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 361-378. http://geodesic.mathdoc.fr/item/SMJ2_2002_28_4_a6/