Nonlinear Quartic Quasi-interpolant Splines to Approximate Piecewise Smooth Functions

Quasi-interpolation based on spline approximation methods is used in numerous applications. A quartic quasi-interpolating spline ([2]) is a piecewise polynomial of degree four satisfying C3 continuity and five order of 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 talk, we present nonlinear modifications of such a spline, based on weighted essentially non-oscillatory (WENO) techniques ([1]) to avoid this phenomena near discontinuities and, at the same time, maintain the five order accuracy in smooth regions. We also provide some numerical and graphical tests confirming the theoretical results.