Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

Abstract

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ${\mathrm{ℒ}}_m^{\text{AG}}f$ has the property of $m\left(m\in \mathbb{Z},m>0\right)$ degree polynomial reproducing and converges up to a rate of $m+1$ . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.