  Analytic continuation problems are notoriously ill-posed without additional
regularizing constraints, even though every analytic function has a rigidity
property of unique continuation from every curve inside the domain of
analyticity. In fact, well known theorems, guarantee that every continuous
function can be uniformly approximated by analytic functions (polynomials or
rational functions, for example). We consider several analytic continuation
problems with typical global boundedness constraints. All such problems exhibit
a power law precision deterioration as one moves away from the source of data.
In this paper we demonstrate the effectiveness of our general Hilbert
space-based approach for determining these exponents. The method identifies the
`worst case' function as a solution of a linear equation with a compact
operator. In special geometries, such as the circular annulus this equation can
be solved explicitly. The obtained solution is then used to determine the power
law exponent for the analytic continuation from an interval between the foci of
a Bernstein ellipse to the entire ellipse. In those cases where such exponents
have been determined in prior work our results reproduce them faithfully.
