  We formalize the use of projections onto convex sets (POCS) for the
reconstruction of signals from non-uniform samples in their highest generality.
This covers signals in any Hilbert space $\mathscr H$, including
multi-dimensional and multi-channel signals, and samples that are most
generally inner products of the signals with given kernel functions in
$\mathscr H$. An attractive feature of the POCS method is the unconditional
convergence of its iterates to an estimate that is consistent with the samples
of the input, even when these samples are of very heterogeneous nature on top
of their non-uniformity, and/or under insufficient sampling. Moreover, the
error of the iterates is systematically monotonically decreasing, and their
limit retrieves the input signal whenever the samples are uniquely
characteristic of this signal. In the second part of the paper, we focus on the
case where the sampling kernel functions are orthogonal in $\mathscr H$, while
the input may be confined in a smaller closed space $\mathscr A$ (of
bandlimitation for example). This covers the increasingly popular application
of time encoding by integration, including multi-channel encoding. We push the
analysis of the POCS method in this case by giving a special parallelized
version of it, showing its connection with the pseudo-inversion of the linear
operator defined by the samples, and giving a multiplierless discrete-time
implementation of it that paradoxically accelerates the convergence of the
iteration.
