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

A kind of Abel–Goncharov type operators is surveyed. +e presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. +e construction of new quasi-interpolantsLAG m 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.


Introduction
Assume that f is a function defined on a domain [a, b] ⊂ R containing X and x i ∈ X, i � 0, . . . , N is some distinct points, where has the form where X(·) is an interpolation kernel. Radial basis functions have been employed to solve the above interpolation problems (2) and (3) by many research fellows. Multiquadrics first introduced by Hardy [1], are particularly interesting, on account of their particular convergence property [2,3]. In this study, we denote the multiquadrics and their shape-preserving parameter in (4) by notations ϕ j (·) and c, respectively. By means of accuracy, efficiency, and easy implementation, Franke [4] investigated that multiquadric interpolation is considered to be one of the most schemes in 29 interpolation methods. Based on distinct nodes x j N j�0 , Micchelli [5] proved that multiquadric interpolation is always solvable, but the resulting matrix in interpolation problems (2) and (3) quickly becomes illconditioned with the increase of the nodes. e well-known quasi-interpolation as a weaker form of (3) reproduces all polynomials of degree ≤m, that is, where P m � p: deg(p) ≤ m . In this study, we will apply the quasi-interpolation scheme to overcome the ill-conditioning problem. Beatson and Powell [6] first constructed a univariate quasi-interpolant L B which reproduces constants. Wu and Schaback [7] introduced another quasi-interpolant L D possessing shape-preserving and linear-reproducing properties.
ey proved that the error of the operator L D is O(h 2 |ln h|) when the shape parameter c � O(h) and h � max 1≤j≤n |x j − x j− 1 | . Based on the operator L D in [7], Ling [8] provided a multilevel quasi-interpolant and showed that its convergence rate is O(h 2.5 |ln h|) as c � O(h). To increase the degree of the multiquadric quasi-interpolation operator, Feng and Zhou [9] provided a kind of multiquadric quasi-interpolants, and the operators could have any degree of exactness. At the same time, by applying the operator L B with Hermite interpolation polynomials, Wang et al. [10] proposed a kind of improved quasi-interpolation operators L H 2m− 1 and gave the desired orders of convergence. By combining the operator L B with Lidstone interpolating polynomials [11][12][13], Wu et al. [14] proposed a kind of Lidstone-type multiquadric quasi-interpolants L Λ n possessing any degree of polynomial reproducibility. e authors have given that the approximating capacity of the operator L Λ n is comparable with that of the operator L H 2m− 1 . Furthermore, many researchers applied multiquadric quasiinterpolants to solve differential equations [15][16][17][18][19][20][21][22][23][24][25][26]. Meanwhile, Ali et al. [27] constructed the SDI using Timmer triangular patches, which are used to visualize the energy data, i.e., spatial interpolation in visualizing rainfall data.
By the means of construction idea in [10], we provide a kind of Abel-Goncharov type multiquadric quasi-interpolants by combining the operator L B with Abel-Goncharov interpolating polynomials. e presented operators could reproduce polynomials of higher degree than L B . Under the suitable assumption of shape-preserving parameter c, we obtain the convergence rates of higher order. erefore, we could derive the desired precision of the our operators with an optimal value of c.
e remaining organization of this study is arranged as follows. In Section 2, we give the definition of univariate Abel-Goncharov interpolation polynomials and derive three useful theorems for the error of approximation. Section 3 is devoted to construct Abel-Goncharov type multiquadric quasi-interpolants and study their approximation orders. In Section 4, numerical experiments are shown to compare the approximation capacity of our operators with that of Wang et al.'s quasi-interpolants. Finally, conclusion is given in Section 5.

Proof. Let us verify that
can reproduce all polynomials of degree no more m. It is easily given that , the Abel-Goncharov interpolation formula is obtained as follows: as acting on the space C m [c, d], where c < a 0 , a m < d. Based on Peano's kernel theorem [32], we provide the following integral expression for the remainder (14).
we consider the following integral representations: where and (·) k + denotes the positive part of the k th power of the argument, such that Proof. First say that in the interpolation polynomial (8) where (18) is provided by using the linear functional where Journal of Mathematics us, the first case of (17) is proved by the above process. e rest of the expressions may be obtained by an analogous manner. □ e following theorem provides the desired bounds.
where ‖ · ‖ ∞ denotes the sup-norm on [c, d], and Proof. If c ≤ x ≤ a 0 , then we have from (17), such that We know that the integrands are of type h(t)f (m) (t) with h(t) that does not change sign in [x, a]. By means of the first mean value theorem for integrals, we obtain for some After some calculations, we obtain If t ∈ [a 0 , a 1 ], then By applying the first mean theorem for the above integrals, we have for some (32) After some calculations, we obtain a 1 ,...,a m (x, t) If t ∈ [a 1 , a 2 ], then By applying the first mean theorem for the above integrals, we have for some ξ 2,k ∈ [c, d], k � 2, . . . , m − 1, (35) After some calculations, we obtain In the same way, we have, if t ∈ [a 2 , a 3 ], then for By definition of (19), K a 0 ,a 1 ,...,a m (x, t) of (18) is zero as a m− 1 ≤ t ≤ a m . Substituting into (26) the left-hand sides of (30), (33), (36), (37), (38), and (39) by their respective righthand sides, we finally get the expression as follows: In order to obtain the desired bounds (24), we have the following inequation for k � 0, 1, m: where the inequality follows from (10):

Journal of Mathematics
Finally, we have, after some calculations, Similarly, the remaining expressions of (24) could be proved.

□
Since the algebraic degree of exactness of the operator P m [f; a 0 , a 1 , . . . , a m ] is equal to m, the following result can be given in an analogous manner.

A Kind of Abel-Goncharov Type Multiquadric Quasi-
where x N+j � x N− m+j− 1 , j � 1, . . . , m. By using each set of nodes X im , i � 0, 1, . . . , N, the Abel-Goncharov interpolation operator in (8) can be denoted by P i m , i � 0, 1, . . . , N, i.e., where we have with the Goncharov polynomials Proof. e proof is given similarly to that of eorem 1. □ Furthermore, we recall the well-known multiquadric quasi-interpolation operator L B , defined by Beatson and Powell in [6], as follows: Journal of Mathematics By combining the quasi-interpolation operator L B with Abel-Goncharov interpolation polynomials, we construct a kind of Abel-Goncharov type multiquadric quasi-interpolation operator L AG m as follows: where P i m [f; x i , x i+1 , . . . , x i+m ](x), i � 0, 1, . . . , N, is the Abel-Goncharov interpolation polynomials defined in (47) and x N+j � x N− m+j− 1 , j � 1, 2, . . . , m.

Theorem 6.
e operator L AG m f reproduces all univariate polynomials of degree ≤ m.
Proof. L AG m p � p is proved from the following result: for i � 0, 1, . . . , N.
2. e Convergence Rate of the Operator. In order to give the convergence rate of the operator L AG m , we apply the following notations: for X � x 0 , x 1 , . . . , x N , where #(·) denotes the cardinality function.
us, we can obtain 2r � max |x 1 − x 0 |, |x 2 − x 1 |, . . . , |x N − x N− 1 |}, and M denotes the maximum number of points from X contained in an interval I r (x). en, we provide the following error estimates.

Theorem 7. Suppose that c satisfies
where D is a positive constant and l is a positive integer. If f(x) ∈ C m (I), then and K is a positive constant independent of x and X.

Conclusions
In this study, by combing multiquadric quasi-interpolant L B with the Abel-Goncharov univariate operator, we construct a kind of Abel-Goncharov multiquadric quasiinterpolants L AG m f. Meanwhile, we have also proven that the operators L AG m f possess the m th degree polynomial reproduction property and good convergence capacity, so that it is convenient for people in various applications. Moreover, the associated algorithm is easily implemented.
In our future work, the univariate Abel-Goncharov type multiquadric quasi-interpolants can be extended to the multivariate case. Moreover, we can also apply the operators to fit scattered data.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.