Nonlinear quartic quasi-interpolating splines for piecewise smooth function approximation

Quasi-interpolation based on spline approximation methods is used in numerous applications. A quartic quasi-interpolating spline is a piecewise polynomial of degree four satisfying C3 continuity and fifth-order approximation, if the function to be approximated is sufficiently smooth. However, if the function has jump discontinuities, we observe that the Gibbs phenomenon appears when approximating near discontinuities. In this paper, we present nonlinear modifications of such a spline, based on weighted essentially non-oscillatory (WENO) techniques to avoid this phenomenon near discontinuities and, at the same time, maintain the fifth-order accuracy in smooth regions. We also provide some numerical and graphical tests confirming the theoretical results.