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Title of article :
On the Degree of Multivariate Bernstein Polynomial Operators Original Research Article
Author/Authors :
C.K. Chui، نويسنده , , D. Hong، نويسنده , , S.T. Wu، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 1994
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Abstract :
Let σ be a d-dimensional simplex with vertices v0, ..., vd and Bn(ƒ,·) denote the nth degree Bernstein polynomial of a continuous function ƒ on σ. Dahmen and Micchelli (Stud. Sci. Hungar.23 (1988), 265-287) proved that Bn(ƒ,·) ≥ Bn+1(ƒ,·), n ∈ N, for any convex function ƒ on σ, and it is clear that a necessary and sufficient condition for the inequality to become an identity for all n ∈ N is that ƒ is an affine polynomial. Let σm be the mth simplicial subdivision of σ (which will be defined precisely later). By using a degree-raising formula, the result of Dahmen and Micchelli can be extended to Bmn(ƒ,·), ≥ Bmn+1(ƒ,·), n ∈ N, for any which is convex on every cell of σm. The objective of this paper is to derive conditions under which this inequality becomes an identity.
Journal title :
Journal of Approximation Theory
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