  We consider approximating analytic functions on the interval $[-1,1]$ from
their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen
\& Kuijlaars states that fast and stable approximation from equispaced samples
is generally impossible. In particular, any method that converges exponentially
fast must also be exponentially ill-conditioned. We prove a positive
counterpart to this `impossibility' theorem. Our `possibility' theorem shows
that there is a well-conditioned method that provides exponential decay of the
error down to a finite, but user-controlled tolerance $\epsilon > 0$, which in
practice can be chosen close to machine epsilon. The method is known as
\textit{polynomial frame} approximation or \textit{polynomial extensions}. It
uses algebraic polynomials of degree $n$ on an extended interval
$[-\gamma,\gamma]$, $\gamma > 1$, to construct an approximation on $[-1,1]$ via
a SVD-regularized least-squares fit. A key step in the proof of our main
theorem is a new result on the maximal behaviour of a polynomial of degree $n$
on $[-1,1]$ that is simultaneously bounded by one at a set of $m+1$ equispaced
nodes in $[-1,1]$ and $1/\epsilon$ on the extended interval $[-\gamma,\gamma]$.
We show that linear oversampling, i.e., $m = c n \log(1/\epsilon) /
\sqrt{\gamma^2-1}$, is sufficient for uniform boundedness of any such
polynomial on $[-1,1]$. This result aside, we also prove an extended
impossibility theorem, which shows that such a possibility theorem (and
consequently the method of polynomial frame approximation) is essentially
optimal.

Numerical Analysis;Numerical Analysis