论文标题
通过hermite插值的多项式近似的确切次数估计值
Exact pointwise estimates for polynomial approximation with Hermite interpolation
论文作者
论文摘要
我们通过有限的多项式（Hermite）插值条件以有限的多项式建立了最佳的点（达到恒定倍数）估计值，以在有限的间隔内建立近似值，并表明这些估计值无法改善。 In particular, we show that {\bf any} algebraic polynomial of degree $n$ approximating a function $f\in C^r(I)$, $I=[-1,1]$, at the classical pointwise rate $ρ_n^r(x) ω_k(f^{(r)}, ρ_n(x))$, where $ρ_n（x）= n^{ - 1} \ sqrt {1-x^2}+n^{ - 2} $，以及（Hermite）interpolating $ f $及其派生词及其衍生物及其派生$ r $在i $中的点$ x_0 \，在i $中是$ x_0 \，具有最佳的点率（同步）$ $ $ $ $ $ $ $ $ $ $ f。 给出了几个申请。
We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that {\bf any} algebraic polynomial of degree $n$ approximating a function $f\in C^r(I)$, $I=[-1,1]$, at the classical pointwise rate $ρ_n^r(x) ω_k(f^{(r)}, ρ_n(x))$, where $ρ_n(x)=n^{-1}\sqrt{1-x^2}+n^{-2}$, and (Hermite) interpolating $f$ and its derivatives up to the order $r$ at a point $x_0\in I$, has the best possible pointwise rate of (simultaneous) approximation of $f$ near $x_0$. Several applications are given.
