Approximation of piecewise smooth functions by nonuniform nonlinear quadratic and cubic spline quasi-interpolants

This paper presents new nonlinear quadratic and cubic spline quasi-interpolants for approximating piecewise smooth functions on nonuniform knot partitions. By incorporating WENO techniques in the quasi-interpolant definition, our method avoids Gibbs phenomenon near discontinuities while maintaining high-order accuracy in smooth regions. The construction extends previous research by considering nonuniform knot distributions, offering enhanced flexibility in function reconstruction. Numerical experiments validate the method’s superior performance compared to linear approaches.