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 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.
Highlights
Assume that f is a function defined on a domain [a, b] ⊂ R containing X and xi ∈ X, i 0, . . . , N is some distinct points, where a x0 < · · · < xN b, (1) has the form NL[f; a, b](x) λjXx − xj, (2) j 0 s.t.L[f; a, b]xj fxj, for j 0, 1, . . . , N, (3)where X(·) is an interpolation kernel
By the means of construction idea in [10], we provide a kind of Abel–Goncharov type multiquadric quasi-interpolants by combining the operator LB with Abel–Goncharov interpolating polynomials. e presented operators could reproduce polynomials of higher degree than LB
By combining the quasi-interpolation operator LB with Abel–Goncharov interpolation polynomials, we construct a kind of Abel–Goncharov type multiquadric quasi-interpolation operator LAmG as follows: N
Summary
Assume that f is a function defined on a domain [a, b] ⊂ R containing X and xi ∈ X, i 0, . . . , N is some distinct points, where a x0 < · · · < xN b,. Wu and Schaback [7] introduced another quasi-interpolant LD possessing shape-preserving and linear-reproducing properties. By applying the operator LB with Hermite interpolation polynomials, Wang et al [10] proposed a kind of improved quasi-interpolation operators LH2m− 1 and gave the desired orders of convergence. By combining the operator LB with Lidstone interpolating polynomials [11,12,13], Wu et al [14] proposed a kind of Lidstone-type multiquadric quasi-interpolants L Λn possessing any degree of polynomial reproducibility. By the means of construction idea in [10], we provide a kind of Abel–Goncharov type multiquadric quasi-interpolants by combining the operator LB with Abel–Goncharov interpolating polynomials.
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