##### A Fourier continuation framework for high-order approximations
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 certain Fourier continuation (FC) based approximation scheme has been demonstrated to be an effective alternative. However, while the practical efficacy of FC based schemes in obtaining a high-order numerical solution of Partial Differential Equations is known, theoretical convergence analyses largely remain unavailable. In this paper, we present a Fourier continuation framework for eliminating Gibbs oscillations from approximations and include an associated convergence analysis. Moreover, we suggest an explicit 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.
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MathPaper: A Fourier continuation framework for high-order approximations. https://t.co/romtCUlnL3