  We consider the problem of approximating an analytic function on a compact
interval from its values at $M+1$ distinct points. When the points are
equispaced, a recent result (the so-called impossibility theorem) has shown
that the best possible convergence rate of a stable method is root-exponential
in $M$, and that any method with faster exponential convergence must also be
exponentially ill-conditioned at a certain rate. This result hinges on a
classical theorem of Coppersmith & Rivlin concerning the maximal behaviour of
polynomials bounded on an equispaced grid. In this paper, we first generalize
this theorem to arbitrary point distributions. We then present an extension of
the impossibility theorem valid for general nonequispaced points, and apply it
to the case of points that are equidistributed with respect to (modified)
Jacobi weight functions. This leads to a necessary sampling rate for stable
approximation from such points. We prove that this rate is also sufficient, and
therefore exactly quantify (up to constants) the precise sampling rate for
approximating analytic functions from such node distributions with stable
methods. Numerical results -- based on computing the maximal polynomial via a
variant of the classical Remez algorithm -- confirm our main theorems. Finally,
we discuss the implications of our results for polynomial least-squares
approximations. In particular, we theoretically confirm the well-known
heuristic that stable least-squares approximation using polynomials of degree
$N < M$ is possible only once $M$ is sufficiently large for there to be a
subset of $N$ of the nodes that mimic the behaviour of the $N$th set of
Chebyshev nodes.
