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 ℒ m AG f has the property of m m ∈ ℤ , m > 0 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.

In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate quasi-interpolation operators only demand information of the location points rather than the derivatives of the function approximated. Some error bounds in terms of the modulus of continuity of high order and Peano representations for the error are given. Several numerical comparisons with other existing methods are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.

It is known that an efficient method for interpolation of very large scattered data sets is the method of Shepard. Unfortunately, it reproduces only the constants. In this paper, we first generalize an expansion in bivariate even order Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Finally, by combining the known Shepard operator with the even order Bernoulli bivariate operator, we construct a kind of new approximated operator satisfying the higher order polynomial reproducibility. We study this combined operator and give some error bounds in terms of the modulus of continuity of high order and also with Peano's theorem. Numerical comparisons show that this new technique provides the higher degree of accuracy. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.

Multiquadric quasi-interpolation is an efficient high-dimensional approximation algorithm. It can directly obtain the approximation term and its derivatives without solving any large-scale linear equations. Firstly, we study a kind of bivariate even order Bernoulli-type multiquadric quasi-interpolation operator by combining the extended univariate known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate even order Bernoulli polynomials. Secondly, we modify a kind of multiquadric quasi-interpolation operator with the property of high degree polynomial reproducing to gridded data on multi-dimensional spaces by using multivariate divided difference to approximate the partial derivatives. Thirdly, error bounds of the modulus of continuity of high order and Peano representation for the error are given. Numerical experiments verity the effectiveness and high accuracy of the scheme. Furthermore, applying the proposed quasi-interpolation operator, we present the explicit algorithm of numerical differentiation. Several examples to the fitting of discrete solutions of two-dimensional coupled Burgers’ equations are given.