Abstract
Suppose that we want to approximate f∈C[0,1] by polynomials inP, using only its values on Xn={i/n, 0≤i≤n}. This can be done by the Lagrange interpolant Ln f or the classical Bernstein polynomial Bn f. But, when n tends to infinity, Ln f does not converge to f in general and the convergence of Bn f to f is very slow. We define a family of operators B (k)n , n≥k, which are intermediate ones between B (0)n =B (1)n =Bn and B (n)n =Ln, and we study some of their properties. In particular, we prove a Voronovskaja-type theorem which asserts that B (k)n f−f=O(n−[(k+2)/2]) for f sufficiently regular.
Moreover, B (k)n f uses only values of Bn f and its derivaties and can be computed by De Casteljau or subdivision algorithms.
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Sablonniere, P. A family of Bernstein quasi-interpolants on [0,1]. Approx. Theory & its Appl. 8, 62–76 (1992). https://doi.org/10.1007/BF02836339
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DOI: https://doi.org/10.1007/BF02836339