We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C^∞ function f defined on a convex open subset Ω⊂R^d containing the d-dimensional simplex S^d of R^d. Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on S^d but also on the whole Ω. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja’s formula is also stated.
Extrapolation Properties of Multivariate Bernstein Polynomials
Campiti M.
;
2019-01-01
Abstract
We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C^∞ function f defined on a convex open subset Ω⊂R^d containing the d-dimensional simplex S^d of R^d. Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on S^d but also on the whole Ω. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja’s formula is also stated.File in questo prodotto:
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