Robust and Consistent Sampling

We address a sampling problem in which the goal is to approximate a signal from its nonideal (generalized) samples. The reconstruction is constrained to lie in a subspace, and to be consistent with the measured samples. It is well known how to obtain a consistent approximation, if the sampling and reconstruction spaces satisfy a certain direct sum condition. Here we show when consistency can be achieved without the need for this condition. The proposed solution provides geometrical insight into the structure of the problem and extends previous treatments by finding a consistent and simultaneously robust approximation of the signal from its samples.