  Identifying holomorphic functions by moment conditions is a classical
problem. This paper explores ideas introduced by Tumanov in a well known paper
on the strip problem, and building on the author's previous work, to show that
weighted Bergman projections can be computed via asymptotic formulas involving
Szeg\H{o} projections a fixed 1-dimensional family of curves. Various new
operators are introduced; in addition, similar formulas are derived via
dimension reduction, using the author's CR Hartogs theorem.
