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Some properties of bivariate Bernstein-Schoenberg operators

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The bivariate Bernstein-Schoenberg operatorV T of degreem, introduced in [5], is a spline approximation operator that generalizes the Bernstein polynomial operatorB m . It is shown here that for a convex functionf,fV T (f)≤B m (f). This result is then used to show that for a twice differentiable functiong, the asymptotic error limm(V T (g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances thatV T is, in a sense, asymptotically close toB m .

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Communicated by Klaus Höllig.

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Goodman, T.N.T. Some properties of bivariate Bernstein-Schoenberg operators. Constr. Approx 3, 123–130 (1987). https://doi.org/10.1007/BF01890558

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Key words and phrases

  • Bernstein-Schoenberg operator
  • Bernstein polynomial
  • MultivariateB-spline
  • Asymptotic error

AMS classification

  • 41A15
  • 41A63