  Under investigation is the problem of finding the best approximation of a
function in a Hilbert space subject to convex constraints and prescribed
nonlinear transformations. We show that in many instances these prescriptions
can be represented using firmly nonexpansive operators, even when the original
observation process is discontinuous. The proposed framework thus captures a
large body of classical and contemporary best approximation problems arising in
areas such as harmonic analysis, statistics, interpolation theory, and signal
processing. The resulting problem is recast in terms of a common fixed point
problem and solved with a new block-iterative algorithm that features
approximate projections onto the individual sets as well as an extrapolated
relaxation scheme that exploits the possible presence of affine constraints. A
numerical application to signal recovery is demonstrated.
