  This paper establishes error bounds for the convergence of a piecewise linear
approximation of the constrained optimal smoothing problem posed in a
reproducing kernel Hilbert space (RKHS). This problem can be reformulated as a
Bayesian estimation problem involving a Gaussian process related to the kernel
of the RKHS. Consequently, error bounds can be interpreted as a quantification
of the maximum a posteriori (MAP) accuracy. To our knowledge, no error bounds
have been proposed for this type of problem so far. The convergence results are
provided as a function of the grid size, the regularity of the kernel, and the
distance from the kernel interpolant of the approximation to the set of
constraints. Inspired by the MaxMod algorithm from recent literature, which
sequentially allocates knots for the piecewise linear approximation, we conduct
our analysis for non-equispaced knots. These knots are even allowed to be
non-dense, which impacts the definition of the optimal smoothing solution and
our error bound quantifiers. Finally, we illustrate our theorems through
several numerical experiments involving constraints such as boundedness and
monotonicity.
