  In this paper, we consider sampling and reconstruction of signals in a
reproducing kernel subspace of $L^p(\Rd), 1\le p\le \infty$, associated with an
idempotent integral operator whose kernel has certain off-diagonal decay and
regularity. The space of $p$-integrable non-uniform splines and the
shift-invariant spaces generated by finitely many localized functions are our
model examples of such reproducing kernel subspaces of $L^p(\Rd)$. We show that
a signal in such reproducing kernel subspaces can be reconstructed in a stable
way from its samples taken on a relatively-separated set with sufficiently
small gap. We also study the exponential convergence, consistency, and the
asymptotic pointwise error estimate of the iterative approximation-projection
algorithm and the iterative frame algorithm for reconstructing a signal in
those reproducing kernel spaces from its samples with sufficiently small gap.
