  Contrary to the traditional pursuit of research on nonuniform sampling of
bandlimited signals, the objective of the present paper is not to find sampling
conditions that permit perfect reconstruction, but to perform the best possible
signal recovery from any given set of nonuniform samples, whether it is finite
as in practice, or infinite to achieve the possibility of unique reconstruction
in $L^2({\mathbb R})$. This leads us to consider the pseudo-inverse of the
whole sampling map as a linear operator of Hilbert spaces. We propose in this
paper an iterative algorithm that systematically performs this pseudo-inversion
under the following conditions: (i) the input lies in some closed space
$\mathscr A$ (such as a space of bandlimited functions); (ii) the samples are
formed by inner product of the input with given kernel functions; (iii) these
functions are orthogonal at least in a Hilbert space $\mathscr H$ that contains
$\mathscr A$. This situation turns out to appear in certain time encoders that
are part of the increasingly important area of event-based sampling. As a
result of pseudo-inversion, we systematically achieve perfect reconstruction
whenever the samples uniquely characterize the input, we obtain minimal-norm
estimates when the sampling is insufficient, and the reconstruction errors are
controlled in the case of noisy sampling. The algorithm consists in alternating
two projections according to the general method of projections onto convex sets
(POCS) and can be implemented by iterating time-varying discrete-time
filtering. We finally show that our signal and sampling assumptions appear in a
nontrivial manner in other existing problems of data acquisition. This includes
multi-channel time encoding where $\mathscr H$ is of the type $L^2({\mathbb
R})^M$, and traditional point sampling with the adoption of a Sobolev space
$\mathscr H$.
