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Complete asymptotic expansion for multivariate Bernstein-Durrmeyer operators and quasi-interpolants. (English) Zbl 1189.41012

Bernstein-Durrmeyer operators for the one dimensional unweighted case were introduced around 1970 and extended to Jacobi-weights in a multi dimensional setting by many authors (for some good historical information the reader can consult the paper under review!).
After stating several known results (named Theorems A, B and C), the main theorem asserts a complete asymptotic expansion for the operator \(\mathbf{M}_{n,\mu}\) in terms of operators \(\mathbf{U}_{k,\ell}\) as \(n\) tends to infinity:
Let \(x\in\mathbf{S}^d\) and \(q\in\mathbf{N}\). Assume that a function \(f\) is bounded in \(\mathbf{S}^d\), all its partial derivatives \(D^{\sigma}f\) with \(|\sigma|\leq 2q\) exist in a certain neighborhood of \(x\) and are continuous at \(x\). Then \[ (\mathbf{M}_{n,\mu}f)(x)=f(x)+\sum_{k=1}^q\, {(-1)^k\over {n+d+|\mu|+k \choose k} } (\mathbf{U}_{k,\mu}f)(x)+o(n^{-q}),\;n\rightarrow\infty. \]
Furthermore, in section 3, it is shown that the complete asymptotic expansion of the derivatives of \(\mathbf{M}_{n,\mu}f\) can be obtained by term-by-term differentiation of the result above.
Finally (section 4) the same type of result is given for the natural quasi-interpolants of the Bernstein-Durrmeyer operators defined by \[ \mathbf{M}_{n,\mu}^{(r)}f:=\sum_{\ell=0}^r\, {1\over{n \choose \ell}} \mathbf{U}_{\ell,\mu}\mathbf{M}_{n,\mu}f. \] These \(\mathbf{M}_{n,\mu}^{(r)}f\) reproduce polynomials of total degree at most \(r\) for \(0\leq r\leq n\).
It is outside the scope of a review of this type to give all the defining formulae; the results stated above are of course clear to people working in the field. This is a well-written paper containing a balanced overview of the historical development of the subject.

MSC:

41A28 Simultaneous approximation
41A05 Interpolation in approximation theory
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References:

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