Asymptotic expansion of the multivariate Bernstein polynomials on a simplex (Q2721747)

For univariate Bernstein polynomials the classical asymptotic expansions include the parameter \(n\) in a quite implicit way. But there are also expansions in terms of powers of the parameter known. The authors derive a similar expansion in the multivariate setting. Using standard multiindex notation multivariate Bernstein polynomials on the simplex are defined as NEWLINE\[NEWLINE B_n(f;x) = \sum_{|k|\leq n}{n\choose k}x^k(1-x)^{n-k} f\left({{k}\over{n}}\right). NEWLINE\]NEWLINE If \(f\) is bounded with continuous partial derivatives of order \(\leq q\), it is shown that NEWLINE\[NEWLINE B_n(f;x)=f(x)+\sum_{k=1}^q n^{-k} \sum_{k<|s|\leq 2k}{{1}\over{s!}} \left({{\partial^{|s|}}\over{\partial x_1^{s_1}\cdots \partial x_d^{s_d}}} f(x)\right) + \sum_{\nu\leq s}a(k,s,\nu)x^{s-\nu}+o(n^{-q}), NEWLINE\]NEWLINE for \(n\to\infty\). The coefficients \(a(k,s,\nu)\) are calculated explicitly. The result allows to deduce a \textit{E. V. Voronovskaya} type result [C. R. Dokl. Acad. Sci. URSS A No. 4, 79-85 (1932; Zbl 0005.01205)] for the operator \(B_n\). Furtheron representations for moments of the Bernstein operator are given.