  It is well known that approximation of functions on $[0,1]$ whose periodic
extension is not continuous fail to converge uniformly due to rapid Gibbs
oscillations near the boundary. Among several approaches that have been
proposed toward the resolution of Gibbs phenomenon in recent years, a Fourier
continuation (FC) based approximation scheme has been suggested by Bruno and
collaborators in the context of certain PDE solvers where approximation grids
used are equispaced. While the practical efficacy of FC based schemes in
obtaining a high-order numerical solution of PDEs is well known, theoretical
convergence analyses largely remain unavailable. The primary objective of this
paper is to take a step in this direction where we analyze the convergence
rates of a Fourier continuation framework for approximations based on discrete
functional data coming from equispaced grids. In this context, we explore a
certain two-point Hermite interpolation strategy for constructing Fourier
continuations that, not only simplifies the implementation of such
approximations but also makes possible a rigorous analysis of its numerical
properties. In particular, we show that the approximations converge with order
$r+1$ for functions coming from a subspace of $C^{r,1}([0,1])$, the space of
$r$-times continuously differentiable function whose $r$th derivative is
Lipschitz continuous. We also demonstrate that theoretical rates are indeed
achieved in practice, through a variety of numerical experiments.
