  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 $\rho_n^r(x) \omega_k(f^{(r)}, \rho_n(x))$, where
$\rho_n(x)=n^{-1}\sqrt{1-x^2}+n^{-2}$, 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.
