摘要
In this note we study the local behaviour of the multi-variate Bernstein polynomials B, on the d-dimensional simplex S?R d. For function f admitting derivatives of sufficient high order in x∈S we derive the complete asymptotic expansion of Bnf as n tends to infinity. All the coefficients of n?1 that only depend on f and x are calculated explicitly. It turns out that combinatorial numbers play an important role. Par result generalize recent formulae due to R. Zhang in a way.
In this note we study the local behaviour of the multi-variate Bernstein polynomials B, on the d-dimensional simplex S?R d. For function f admitting derivatives of sufficient high order in x∈S we derive the complete asymptotic expansion of Bnf as n tends to infinity. All the coefficients of n?1 that only depend on f and x are calculated explicitly. It turns out that combinatorial numbers play an important role. Par result generalize recent formulae due to R. Zhang in a way.
