A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data (Q1006022)

A new univariate quasi-interpolation operator \({\mathcal L}_d:\, f\mapsto{\mathcal L}_d(f;\, x)\) is introduced over the space of continuous real-valued functions. It is defined as span of function values over the nonuniform mesh \(a=x_0<x_1<\dots <x_n=b\), and basis functions made of divided differences of the cubic multiquadric functions \(\phi(x)= (x^2+c^2)^{3/2}\) ( \(c>0\) is the shape parameter), and it does not require derivatives of \(f\). Further, \({\mathcal L}_d\) preserves quadratic polynomials as well as convexity of orders 2 and 3. Its approximation capacity is good, in fact, it satisfies \(\|{\mathcal L}_d(f;\, x)-f(x)\|_{\infty}\leq O(h^3)+O(ch^2)+O(c^2h)+O(c^2)\), \(h=max(x_{i+1}-x_i)\). Some numerical examples are given to show advantages of \({\mathcal L}_d\) over the quasi-interpolation operator \({\mathcal L}_2\), introduced by \textit{Z. Wu} and \textit{R. Schaback} [Acta Math. Appl. Sin., Engl. Ser. 10, No. 4, 441--446 (1994; Zbl 0822.41025)].