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 any algebraic polynomial of degree $n$ approximating a function $f\in C^r\left(I\right)$, $I=[-1,1]$, at the classical pointwise rate $c(k,r){\rho }_n^r\left(x\right){\omega }_k(f^{\left(r\right)},{\rho }_n\left(x\right))$, where ${\rho }_n\left(x\right)=n^{-1}\sqrt{1-x^2}+n^{-2}$, and $c(k,r)$ is a constant which depends only on $k$ and $r$, and is independent of $f$ and $n$; 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.

我们通过满足有限多个（Hermite）插值条件的多项式，在有限的间隔上建立最佳的逐点估计（最多为常数倍），以进行近似，并表明这些估计无法改进。特别是，我们证明了任何度数的代数多项式$ñ$ 近似函数 $F\in C^{\mathrm{[R}}（\mathrm{一世}）$， $\mathrm{一世}=[-\mathrm{1个}，\mathrm{1个}]$，以经典的逐点速率 $C（ķ，\mathrm{[R}）{\rho }_{ñ}^{\mathrm{[R}}（X）{\omega }_{ķ}（F^{（\mathrm{[R}）}，{\rho }_{ñ}（X））$，在哪里 ${\rho }_{ñ}（X）={ñ}^{-\mathrm{1个}}\sqrt{\mathrm{1个}-X^2}+{ñ}^{-2}$和 $C（ķ，\mathrm{[R}）$ 是一个常数，仅取决于 $ķ$ 和 $\mathrm{[R}$，并且独立于 $F$ 和 $ñ$; 和（Hermite）内插$F$ 及其衍生品直至订单 $\mathrm{[R}$ 在某一点上 $X_0\in \mathrm{一世}$，具有（同时）近似的最佳点方向速率 $F$ 近 $X_0$。给出了几种应用。