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 has the property of degree polynomial reproducing and converges up to a rate of . 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.
1. Introduction
Assume that is a function defined on a domain containing and , is some distinct points, wherehas the forms.t.where 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 and , 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 , Micchelli [5] proved that multiquadric interpolation is always solvable, but the resulting matrix in interpolation problems (2) and (3) quickly becomes ill-conditioned with the increase of the nodes. The well-known quasi-interpolation as a weaker form of (3) reproduces all polynomials of degree , that is,where . 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 which reproduces constants. Wu and Schaback [7] introduced another quasi-interpolant possessing shape-preserving and linear-reproducing properties. They proved that the error of the operator is when the shape parameter and . Based on the operator in [7], Ling [8] provided a multilevel quasi-interpolant and showed that its convergence rate is as . 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 with Hermite interpolation polynomials, Wang et al. [10] proposed a kind of improved quasi-interpolation operators and gave the desired orders of convergence. By combining the operator with Lidstone interpolating polynomials [11–13], Wu et al. [14] proposed a kind of Lidstone-type multiquadric quasi-interpolants possessing any degree of polynomial reproducibility. The authors have given that the approximating capacity of the operator is comparable with that of the operator . Furthermore, many researchers applied multiquadric quasi-interpolants to solve differential equations [15–26]. 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 with Abel–Goncharov interpolating polynomials. The presented operators could reproduce polynomials of higher degree than . Under the suitable assumption of shape-preserving parameter , we obtain the convergence rates of higher order. Therefore, we could derive the desired precision of the our operators with an optimal value of .
The 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.
2. Univariate Abel–Goncharov Interpolation Polynomial
We recall first Abel–Goncharov interpolation problem from [28–30]. Consider , , , and suppose is a function with the first derivatives , . For the nodes , , and the values , , we introduce the Abel–Goncharov interpolation problem of existing polynomial of degree , s.t.
The determinant of the linear system,is always nonzero, and problem (6) has a unique solution. The Abel–Goncharov interpolation polynomial could be expressed as follows:where are known as the Goncharov polynomials of degree k [31] with
The Abel–Goncharov interpolation polynomial has the following properties:
Remark 1. As the nodes , the Abel–Goncharov interpolation polynomial is the th Taylor polynomial of about .
The Abel–Goncharov interpolation polynomial has the polynomial reproduction property as follows.
Theorem 1. The Abel–Goncharov interpolation polynomial reproduces all polynomials of degree no more .
Proof. Let us verify that , for . The Abel–Goncharov interpolation polynomial can reproduce all polynomials of degree no more . It is easily given thatwhile
For function , the Abel–Goncharov interpolation formula is obtained as follows:
For bounds of the above remainder even in points outside the interval , we use the operatoras acting on the space , where . Based on Peano’s kernel theorem [32], we provide the following integral expression for the remainder (14).
Theorem 2. Given and , for the remainderwe consider the following integral representations:whereand denotes the positive part of the kth power of the argument, such that
Proof. First say that in the interpolation polynomial (8), there are evaluations of derivatives of function up to the order in the interval . Finally, the approximant (8) has the degree of exactness . By using Peano’s kernel theorem, we then obtainwhere (18) is provided by using the linear functional to viewed as a function of .
If , thenLet , thenwhere is expressed as a polynomial in of degree .
Let , thenThus, the first case of (17) is proved by the above process. The rest of the expressions may be obtained by an analogous manner.
The following theorem provides the desired bounds.
Theorem 3. Given and , for the remainder (16), we havewhere denotes the sup-norm on , and
Proof. If , then we have from (17),If , thensuch thatWe know that the integrands are of type with that does not change sign in . By means of the first mean value theorem for integrals, we obtain for some ,After some calculations, we obtainIf , thenBy applying the first mean theorem for the above integrals, we have for some ,After some calculations, we obtainIf , thenBy applying the first mean theorem for the above integrals, we have for some ,After some calculations, we obtainIn the same way, we have, if , then for ,If , then for ,If , then for ,By definition of (19), of (18) is zero as . Substituting into (26) the left-hand sides of (30), (33), (36), (37), (38), and (39) by their respective right-hand sides, we finally get the expression as follows:In order to obtain the desired bounds (24), we have the following inequation for :where the inequality follows from (10):Finally, we have, after some calculations,Similarly, the remaining expressions of (24) could be proved.
Since the algebraic degree of exactness of the operator is equal to , the following result can be given in an analogous manner.
Theorem 4. Given and , for the remainder (16), we havewhere is defined in (25).
3. The Abel–Goncharov Type Multiquadric Quasi-Interpolants
3.1. A Kind of Abel–Goncharov Type Multiquadric Quasi-Interpolants
For pairwise distinct points , and the values , we attach to each node , a set of nodes , , defined bywhere . By using each set of nodes , the Abel–Goncharov interpolation operator in (8) can be denoted by , i.e.,where we havewith the Goncharov polynomials
Theorem 5. The Abel–Goncharov interpolants , possess the degree of exactness .
Proof. The proof is given similarly to that of Theorem 1.
Furthermore, we recall the well-known multiquadric quasi-interpolation operator , defined by Beatson and Powell in [6], as follows:wherefor , where is the piecewise linear hat function with the knots and satisfies . The quasi-interpolation operator reproduces constants, i.e.,
By combining the quasi-interpolation operator with Abel–Goncharov interpolation polynomials, we construct a kind of Abel–Goncharov type multiquadric quasi-interpolation operator as follows:where , , is the Abel–Goncharov interpolation polynomials defined in (47) and .
Theorem 6. The operator reproduces all univariate polynomials of degree .
Proof. is proved from the following result:for .
3.2. The Convergence Rate of the Operator
In order to give the convergence rate of the operator , we apply the following notations:for , where denotes the cardinality function. Thus, we can obtain denotes the maximum number of points from contained in an interval . Then, we provide the following error estimates.
Theorem 7. Suppose that satisfieswhere is a positive constant and is a positive integer. If , thenwhereand is a positive constant independent of and .
Proof. Let be fixed, , ; for each , we setBased on (24), (47), and (52), we can givewhereAssume thatwhere denotes the covering of with half open intervals. Thus, for every , there exists a unique , such that . Then, we give the following inequalities as :We have from the definition of When , we have, after some calculations,where . When , we get in an analogous mannerWhen , we also getwhere . Then, we havewhere the last inequality follows the form
Case 1. .(i)Let . Then, .(ii)Let . Then, .
Case 2. .(i)Let . Then, .(ii)Let . Then, .
In an analogous manner, we can obtain the desired error estimation as follows:
Theorem 8. Let satisfywhere is a positive constant and is a positive integer. If , thenwhereand is a positive constant independent of and .
4. Numerical Examples
In order to investigate the accuracy of our operators, we use the following functions on the interval [33].
For each function , we will compare the numerical results of our new operator with the known operator [10] as . We consider a uniform grid of 17 points for and , grid of 11 points for , and grid of 8 points for on the interval . In order to obtain the estimation as accurate as possible, we calculated the approximated functions at the points . Tables 1–6 present the mean and maximum errors, computed for different values of parameters and . These results show that the approximations of Abel–Goncharov multiquadric quasi-interpolants are comparable with that of the multiquadric quasi-interpolants .
5. Conclusions
In this study, by combing multiquadric quasi-interpolant with the Abel–Goncharov univariate operator, we construct a kind of Abel–Goncharov multiquadric quasi-interpolants . Meanwhile, we have also proven that the operators possess the mth 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
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the Open Project of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant no. 93K172019K13).