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

Authors: Ruifeng Wu
Journal: Journal of Mathematics
 (2021)

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## 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-interpolantsℒmAGf has the property of mm∈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.


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## Body

## 1. Introduction

Assume thatf is a function defined on a domain a,b⊂ℝ containing X and xi∈X, i=0,…,N is some distinct points, where(1)a=x0<⋯<xN=b,has the form(2)ℒf;a,bx=∑j=0NλjXx−xj,s.t.(3)ℒf;a,bxj=fxj,forj=0,1,…,N,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],(4)ϕjx=ϕx−xj=x−xj2+c2,j=0,1,…,N,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 ϕj⋅ 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 xjj=0N, 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 ≤m, that is,(5)ℒp;a,bx=px,p∈ℙm,x∈ℝ,where ℙm=p:degp≤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 ℒB which reproduces constants. Wu and Schaback [7] introduced another quasi-interpolant ℒD possessing shape-preserving and linear-reproducing properties. They proved that the error of the operator ℒD is Oh2lnh when the shape parameter c=Oh and h=max1≤j≤nxj−xj−1. Based on the operator ℒD in [7], Ling [8] provided a multilevel quasi-interpolant and showed that its convergence rate is Oh2.5lnh as c=Oh. 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 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 [11–13], Wu et al. [14] proposed a kind of Lidstone-type multiquadric quasi-interpolants ℒ˜Λn possessing any degree of polynomial reproducibility. The authors have given that the approximating capacity of the operator ℒ˜Λn is comparable with that of the operator ℒH2m−1. 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 ℒB with Abel–Goncharov interpolating polynomials. The presented operators could reproduce polynomials of higher degree than ℒB. Under the suitable assumption of shape-preserving parameter c, we obtain the convergence rates of higher order. Therefore, we could derive the desired precision of the our operators with an optimal value of c.The remaining organization of this study is arranged as follows. In Section2, 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 m∈ℕ, a0,a1,…,am∈ℝ, a0<a1<⋯<am−1<am, and suppose f:a0,am⟶ℝ is a function with the first m derivatives fi, 0≤i≤m. For the nodes ai∈a0,am, 0≤i≤m, and the values fixi, 0≤i≤m, we introduce the Abel–Goncharov interpolation problem of existing polynomial Pmf;a0,a1,…,amx of degree m, s.t.(6)Pmf;a0,a1,…,amiai=fiai,0≤i≤m.The determinant of the linear system,(7)D=1a0a02⋯a0m01!2a1⋯ma1m−1002!⋯mm−1a2m−2⋮⋮⋮⋮000⋯m!,is always nonzero, and problem (6) has a unique solution. The Abel–Goncharov interpolation polynomial Pmf;a0,a1,…,amx could be expressed as follows:(8)Pmf;a0,a1,…,amx=∑k=0mgkxfkak,where gkx,k=0,1,…,m are known as the Goncharov polynomials of degree k [31] with(9)gksas=0,ifk≠s,gkkx=1.By means of [29–31], we have(10)g0x=1,g1x=x−a0,gkx=∫a0xdt1∫a1t1dt2⋯∫ak−1tk−1dtk=1k!xk−∑j=0k−1gjxkjajk−j,k=2,…,m.The Abel–Goncharov interpolation polynomialPmf;a0,a1,…,amx has the following properties:(11)limam⟶a0Pmf;a0,a1,…,amx=∑k=0mx−a0kk!fka0.Remark 1.
As the nodesa0=a1=⋯=am, the Abel–Goncharov interpolation polynomial Pmf;a0,a1,…,amx is the mth Taylor polynomial of f about a0.The Abel–Goncharov interpolation polynomialPmf;a0,a1,…,amx has the polynomial reproduction property as follows.Theorem 1.
The Abel–Goncharov interpolation polynomialPmf;a0,a1,…,amx reproduces all polynomials of degree no more m.Proof.
Let us verify thatPmf;a0,a1,…,amx=fx, for fx=xi,i=0,1,…,m,x∈a0,am. The Abel–Goncharov interpolation polynomial Pmf;a0,a1,…,amx can reproduce all polynomials of degree no more m. It is easily given that(12)Pmxi;a0,a1,…,amx=xi,i=0,1,…,m,while(13)Pmxm+1;a0,a1,…,amx=g0xa0m+1+g1xm+1a1m+⋯+gmxm+1m,…,2⋅am≠xm+1.For functionf∈Cma0,am, the Abel–Goncharov interpolation formula is obtained as follows:(14)fx=Pmf;a0,a1,…,amx+Rmf;a0,a1,…,amx.For bounds of the above remainderRmf;a0,a1,…,am even in points outside the interval c,d, we use the operator(15)f⟶Pmf;a0,a1,…,am,as acting on the space Cmc,d, where c<a0,am<d. Based on Peano’s kernel theorem [32], we provide the following integral expression for the remainder (14).Theorem 2.
Givenf∈Cmc,d and x∈c,d, for the remainder(16)Rmf;a0,a1,…,amx=fx−Pmf;a0,a1,…,amx,we consider the following integral representations:(17)Rmf;a0,a1,…,amx=1m−1!∫xam−1fmtKa0,a1,…,amx,tdt,c≤x≤a0,1m−1!∫a0am−1fmtKa0,a1,…,amx,tdt,a0≤x≤am−1,1m−1!∫a0xfmtKa0,a1,…,amx,tdt,am−1≤x≤d,where(18)Ka0,a1,…,amx,t=x−t+m−1−∑k=0m−1m−1!m−k−1!gkxak−t+m−k−1,and ⋅+k denotes the positive part of the kth power of the argument, such that(19)s+k=sk,s>0,0,s≤0.Proof.
First say that in the interpolation polynomial (8), there are evaluations of derivatives of function f up to the order m−1 in the interval c,d. Finally, the approximant (8) has the degree of exactness m. By using Peano’s kernel theorem, we then obtain(20)Rmf;a0,a1,…,amx=1m−1!∫cdfmtKa0,a1,…,amx,tdt,where (18) is provided by using the linear functional f⟶Rmf;a0,a1,…,am to x−t+m−1 viewed as a function of x.
Ifc≤x≤a0, then(21)Rmf;a0,a1,…,amx=1m−1!∫cxfmtKa0,a1,…,amx,tdt+1m−1!∫xa0fmtKa0,a1,…,amx,tdt+1m−1!∫a0a1fmtKa0,a1,…,amx,tdt+1m−1!∫a1a2fmtKa0,a1,…,amx,tdt+⋯+1m−1!∫am−2am−1fmtKa0,a1,…,amx,tdt+1m−1!∫am−1dfmtKa0,a1,…,amx,tdt.
Lett∈c,x, then(22)Ka0,a1,…,amx,t=x−tm−1−∑k=0m−1m−1!m−k−1!gkxak−tm−k−1=x−tm−1−x−tm−1=0,where x−tm−1 is expressed as a polynomial in x of degree m−1.
Lett∈am−1,d, then(23)Ka0,a1,…,amx,t=0.
Thus, 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.
Givenf∈Cmc,d and x∈c,d, for the remainder (16), we have(24)Rmf;a0,a1,…,amx≤Cmfm∞am−1−xm,c≤x≤a0,Cmfm∞am−1−a0m,a0≤x≤am−1,Cmfm∞x−a0m,am−1≤x≤d,where ⋅∞ denotes the sup-norm on c,d, and(25)Cm=1+∑k=0m−1+∑k=1m−1+⋯+∑k=m−2m−11m−k!.Proof.
Ifc≤x≤a0, then we have from (17),(26)Rmf;a0,a1,…,amx=1m−1!∫xa0fmtKa0,a1,…,amx,tdt+1m−1!∫a0a1fmtKa0,a1,…,amx,tdt+1m−1!∫a1a2fmtKa0,a1,…,amx,tdt+⋯+1m−1!∫am−2am−1fmtKa0,a1,…,amx,tdt.
Ift∈x,a0, then(27)Ka0,a1,…,amx,t=−∑k=0m−1m−1!m−k−1!gkxak−tm−k−1,such that(28)∫xa0Ka0,a1,…,amx,tfmtdt=−∑k=0m−1m−1!m−k−1!gkx∫xa0ak−tm−k−1fmtdt.
We know that the integrands are of typehtfmt with ht that does not change sign in x,a. By means of the first mean value theorem for integrals, we obtain for some ξ0,k∈c,d,k=0,1,…,m−1,(29)∫xa0Ka0,a1,…,amx,tfmtdt=−∑k=0m−1m−1!m−k−1!gkxfmξ0,k∫xa0ak−tm−k−1dt.
After some calculations, we obtain(30)∫xa0Ka0,a1,…,amx,tfmtdt=−∑k=0m−1m−1!m−k−1!gkxfmξ0,kak−xm−k−ak−a0m−k.
Ift∈a0,a1, then(31)Ka0,a1,…,amx,t=−∑k=1m−1m−1!m−k−1gkxak−tm−k−1fmt,∫a0a1Ka0,a1,…,amx,tfmtdt=−∑k=1m−1m−1!m−k−1!gkx∫a0a1ak−tm−k−1fmtdt.
By applying the first mean theorem for the above integrals, we have for someξ1,k∈c,d,k=1,…,m−1,(32)∫a0a1Ka0,a1,…,amx,tfmtdt=−∑k=1m−1m−1!m−k−1!gkxfmξ1,k∫a0a1ak−tm−k−1dt.
After some calculations, we obtain(33)∫a0a1Ka0,a1,…,amx,tfmtdt=−∑k=1m−1m−1!m−k−1!gkxfmξ1,kak−a0m−k−ak−a1m−k.
Ift∈a1,a2, then(34)Ka0,a1,…,amx,t=−∑k=2m−1m−1!m−k−1!gkxak−tm−k−1fmt,∫a1a2Ka0,a1,…,amx,tfmtdt=−∑k=2m−1m−1!m−k!gkx∫a1a2ak−tm−k−1fmtdt.
By applying the first mean theorem for the above integrals, we have for someξ2,k∈c,d,k=2,…,m−1,(35)∫a1a2Ka0,a1,…,amx,tfmtdt=−∑k=2m−1m−1!m−k−1!gkxfmξ2,k∫a1a2ak−tm−k−1dt.
After some calculations, we obtain(36)∫a1a2Ka0,a1,…,amx,tfmtdt=−∑k=2m−1m−1!m−k!gkxfmξ2,kak−a1m−k−ak−a2m−k.
In the same way, we have, ift∈a2,a3, then for ξ3,k∈c,d,k=3,…,m−1,(37)∫a2a3Ka0,a1,…,amx,tfmtdt=−∑k=3m−1m−1!m−k!gkxfmξ3,kak−a2m−k−ak−a3m−k.
Ift∈am−3,am−2, then for ξm−2,k∈c,d,k=m−2,m−1,(38)∫am−3am−2Ka0,a1,…,amx,tfmtdt=−∑k=m−2m−1m−1!m−k!gkxfmξm−2,kak−am−3m−k−ak−am−2m−k.
Ift∈am−2,am−1, then for ξm−1,k∈c,d,k=m−1,(39)∫am−2am−1Ka0,a1,…,amx,tfmtdt=−∑k=m−1m−1m−1!m−k!gkxfmξm−1,kak−am−2m−k−ak−am−1m−k.
By definition of (19), Ka0,a1,…,amx,t of (18) is zero as am−1≤t≤am. 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:(40)Rmf;a0,a1,…,amx,t=−∑k=0m−11m−k!gkxfmξ0,kak−xm−k−ak−a0m−k−∑k=1m−11m−k!gkxfmξ1,kak−a0m−k−ak−a1m−k−⋯−∑k=m−2m−11m−k!gkxfmξm−2,kak−am−3m−k−ak−am−2m−k−∑k=m−1m−11m−k!gkxfmξm−1,kak−am−2m−k−ak−am−1m−k.
In order to obtain the desired bounds (24), we have the following inequation for k=0,1,m:(41)gkx=∫a0xdt1∫a1t1dt2,…,∫ak−1tk−1dtk≤am−1−xk,where the inequality follows from (10):(42)g0x=1≤am−1−x0,g1x=x−x0≤am−1−x,gkx=∫a0xdt1∫a1t1dt2,…,∫ak−1tk−1dtk≤∫a0xdt1∫a1xdt2,…,∫ak−1xdtk≤x−a0x−a1,…,x−ak−1≤am−1−xk,k=2,3,…,m.
Finally, we have, after some calculations,(43)Rmf;a0,a1,…,am≤fm∞am−1−xm∑k=0m−11m−k!+∑k=1m−11m−k!+⋯+∑k=m−2m−11m−k!+∑k=m−1m−11m−k!≤1+∑k=0m−1+∑k=1m−1+⋯+∑k=m−2m−11m−k!fm∞am−1−xm.
Similarly, the remaining expressions of (24) could be proved.Since the algebraic degree of exactness of the operatorPmf;a0,a1,…,am is equal to m, the following result can be given in an analogous manner.Theorem 4.
Givenf∈Cm+1c,d and x∈c,d, for the remainder (16), we have(44)Rmf;a0,a1,…,amx≤Cm+1fm+1∞am−xm+1,c≤x≤a0,Cm+1fm+1∞am−a0m+1,a0≤x≤am,Cm+1fm+1∞x−a0m+1,am≤x≤d,where Cm is defined in (25).


## 3. The Abel–Goncharov Type Multiquadric Quasi-Interpolants

### 3.1. A Kind of Abel–Goncharov Type Multiquadric Quasi-Interpolants

ForN+1 pairwise distinct points xi∈a,b,0≤i≤N, and the values fixi,i=0,1,…,N, we attach to each node xi,i=0,1,…,N, a set of nodes Xim,m∈ℕ,m∈N, i=0,1,…,N, defined by(45)Xim=xi,xi+1,…,xi+m,i=0,1,…,N,where xN+j=xN−m+j−1,j=1,…,m. By using each set of nodes Xim,i=0,1,…,N, the Abel–Goncharov interpolation operator in (8) can be denoted by Pmi,i=0,1,…,N, i.e.,(46)Pmif;xi,xi+1,…,xi+mk=fkxk,i≤k≤i+m,0≤i≤N,where we have(47)Pmif;xi,xi+1,…,xi+mx=∑j=ii+mgj−iixfj−ixj,i=0,1,…,N,with the Goncharov polynomials(48)g0ix=1,g1ix=x−xi,,…,gkix=1k!xk−∑j=0k−1gjixkj,k=2,…,m.Theorem 5.
The Abel–Goncharov interpolantsPmi,i=0,1,…,N, possess the degree of exactness m.Proof.
The proof is given similarly to that of Theorem1.Furthermore, we recall the well-known multiquadric quasi-interpolation operatorℒB, defined by Beatson and Powell in [6], as follows:(49)ℒBfx=fx0ψ0x+∑i=1N−1fxiψix+fxNψNx,x∈I,where(50)ψ0x=12c2∫−∞x01x−t2+c23/2dt+12c2∫x0x1x1−t/x1−x0x−t2+c23/2dt=12+ϕ1x−ϕ0x2x1−x0,ψNx=12c2∫xN∞1x−t2+c23/2+12c2∫xN−1xNt−xN−1/xN−xN−1x−t2+c23/2dt=12−ϕNx−ϕN−1x2xN−xN−1,ψix=12c2∫xi−1xi+1Bitx−t2+c23/2dt=ϕi+1x−ϕix2xi+1−xi−ϕix−ϕi−1x2xi−xi−1,for i=1,2,…,N−1, where Bit is the piecewise linear hat function with the knots xi−1,xi,xi+1 and satisfies Bixi=1. The quasi-interpolation operator ℒB reproduces constants, i.e.,(51)∑i=0Nψix=1.By combining the quasi-interpolation operatorℒB with Abel–Goncharov interpolation polynomials, we construct a kind of Abel–Goncharov type multiquadric quasi-interpolation operator ℒmAG as follows:(52)ℒmAGfx=∑i=0NψixPmif;xi,xi+1,…,xi+mx,x∈I,where Pmif;xi,xi+1,…,xi+mx, i=0,1,…,N, is the Abel–Goncharov interpolation polynomials defined in (47) and xN+j=xN−m+j−1,j=1,2,…,m.Theorem 6.
The operatorℒmAGf reproduces all univariate polynomials of degree ≤m.Proof.
ℒmAGp=p is proved from the following result:(53)∑i=0Nψix=1,Pmip;xi,xi+1,…,xi+m=p,p∈ℙm,for i=0,1,…,N.


### 3.2. The Convergence Rate of the Operator

In order to give the convergence rate of the operatorℒmAG, we apply the following notations:(54)Iρx=x−ρ,x+ρ,ρ>0,r=infρ>0:∀x∈I,Iρx∩X≠∅,M=maxx∈I#Irx∩X,for X=x0,x1,…,xN, where #⋅ denotes the cardinality function. Thus, we can obtain 2r=maxx1−x0,x2−x1,…,xN−xN−1,andM denotes the maximum number of points from X contained in an interval Irx. Then, we provide the following error estimates.Theorem 7.
Suppose thatc satisfies(55)c≤Drl,where D is a positive constant and l is a positive integer. If fx∈CmI, then(56)ℒmAGfx−fx∞≤KMfm∞El,mr,where(57)El,mr=rlnr,m=1,l=1,r,m=1,l>1,rm,m>1,m≤2l−l,r2l−1,m>1,m>2l−l,and K is a positive constant independent of x and X.Proof.
Letxi,xi+1,…,xi+m∈I be fixed, i=0,1,…,N, m∈ℕ,xi<xi+1,…,xi+m; for each x∈I, we set(58)dxi,xi+1,…,xi+m=xi+m−1−x,x≤xi,xi+m−1−xi,xi≤x≤xi+m−1,x−xi,xi+m−1≤x,dmxi,xi+1,…,xi+m=dxi,xi+1,…,xi+mm.
Based on (24), (47), and (52), we can give(59)LmAGfx−fx=∑i=0NψixPmif;xi,xi+1,…,xi+mx−fx≤∑i=0NψixPmif;xi,xi+1,…,xi+mx−fx≤Cmfm∞Sl,mx,where(60)Sl,mx=∑i=0Nψixdmxi,xi+1,…,xi+mx.
Assume that(61)n=xN−x02r+1,Qρu=u−ρ,u+ρ,u∈I,ρ>0,Tj=Qrx−2rj∪Qrx+2rj,j=0,…,n,where ∪j=−nnQrx+2rj denotes the covering of I with half open intervals. Thus, for every i∈0,1,…,N, there exists a unique j∈0,1,…,n, such that xi∈Tj. Then, we give the following inequalities as j≥2:(62)2j−1r≤x−xi≤2j+1r,2j−1−1r≤x−τi≤2j+1+1r,forτi∈xi−1,xi+1,dxi,xi+1,…,xi+mx≤2j+m−1+1r.
We have from the definition ofM(63)1≤#X∩T0≤M,1≤#X∩Tj≤2M,j=1,2,…,n.
Whenx0∈Tj,j=2,3,…,n, we have, after some calculations,(64)ψ0x≤12c2∫−∞x01x−t3dt+12c21x−τ02+c23/2∫x0x1x1−tx1−x0dt≤14c2x−x0−2+14c2x1−x0x−τ0−3≤14c22j−1−2r−2+22j−3−3r−2≤c2r−22j−3−2,where τ0∈x0,x1. When xN∈Tj,j=2,3,…,n, we get in an analogous manner(65)ψNx≤c2r−22j−3−2.When xii=1,…,N−1∈Tj,j=2,3,…,n, we also get(66)ψix≤12c21x−τi2+c23/2∫xi−1xi+1Bitdt≤14c2xi+1−xi−1x−τi−3≤c2r−22j−3−2,where τi∈xi−1,xi+1. Then, we have(67)Sl,mx≤∑xi∈T0,T1ψixdmxi,xi+1,…,xi+mx+∑j=2n∑xi∈Tjψixdmxi,xi+1,…,xi+mx≤∑xi∈T0,T1dmxi,xi+1,…,xi+mx+∑j=2n∑xi∈Tjψixdmxi,xi+1,…,xi+mx≤M2m−1rm+2M2m+1rm+2M∑j=2nc2r−22j−3−22j+2m−1rm≤2M2m+1rm+2m+1rm+c2rm−2∑j=2n2j−3−22j+2m−1m≤2M2m+1m2rm+D2rm+2l−2∑j=1njm−2,where the last inequality follows the form(68)2j−3≥j,j=3,4,…,n,2j+2m−1≤2m+1j,j=1,2,…,n,c≤Drl.Case 1.
m=1.(i)
Letl=1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Orlnr.(ii)
Letl>1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Or.Case 2.
m>1.(i)
Letm≤2l−1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Orm.(ii)
Letm>2l−1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Or2l−1.In an analogous manner, we can obtain the desired error estimation as follows:Theorem 8.
Letc satisfy(69)c≤Drl,where D is a positive constant and l is a positive integer. If fx∈Cm+1I, then(70)ℒmAGfx−fx∞≤K′Mfm+1∞El,m′r,where(71)El,m′r=r2l−1,m+1≥2l−1,rm+1,m+1≤2l−1,and K′ is a positive constant independent of x and X.


## 3.1. A Kind of Abel–Goncharov Type Multiquadric Quasi-Interpolants

ForN+1 pairwise distinct points xi∈a,b,0≤i≤N, and the values fixi,i=0,1,…,N, we attach to each node xi,i=0,1,…,N, a set of nodes Xim,m∈ℕ,m∈N, i=0,1,…,N, defined by(45)Xim=xi,xi+1,…,xi+m,i=0,1,…,N,where xN+j=xN−m+j−1,j=1,…,m. By using each set of nodes Xim,i=0,1,…,N, the Abel–Goncharov interpolation operator in (8) can be denoted by Pmi,i=0,1,…,N, i.e.,(46)Pmif;xi,xi+1,…,xi+mk=fkxk,i≤k≤i+m,0≤i≤N,where we have(47)Pmif;xi,xi+1,…,xi+mx=∑j=ii+mgj−iixfj−ixj,i=0,1,…,N,with the Goncharov polynomials(48)g0ix=1,g1ix=x−xi,,…,gkix=1k!xk−∑j=0k−1gjixkj,k=2,…,m.Theorem 5.
The Abel–Goncharov interpolantsPmi,i=0,1,…,N, possess the degree of exactness m.Proof.
The proof is given similarly to that of Theorem1.Furthermore, we recall the well-known multiquadric quasi-interpolation operatorℒB, defined by Beatson and Powell in [6], as follows:(49)ℒBfx=fx0ψ0x+∑i=1N−1fxiψix+fxNψNx,x∈I,where(50)ψ0x=12c2∫−∞x01x−t2+c23/2dt+12c2∫x0x1x1−t/x1−x0x−t2+c23/2dt=12+ϕ1x−ϕ0x2x1−x0,ψNx=12c2∫xN∞1x−t2+c23/2+12c2∫xN−1xNt−xN−1/xN−xN−1x−t2+c23/2dt=12−ϕNx−ϕN−1x2xN−xN−1,ψix=12c2∫xi−1xi+1Bitx−t2+c23/2dt=ϕi+1x−ϕix2xi+1−xi−ϕix−ϕi−1x2xi−xi−1,for i=1,2,…,N−1, where Bit is the piecewise linear hat function with the knots xi−1,xi,xi+1 and satisfies Bixi=1. The quasi-interpolation operator ℒB reproduces constants, i.e.,(51)∑i=0Nψix=1.By combining the quasi-interpolation operatorℒB with Abel–Goncharov interpolation polynomials, we construct a kind of Abel–Goncharov type multiquadric quasi-interpolation operator ℒmAG as follows:(52)ℒmAGfx=∑i=0NψixPmif;xi,xi+1,…,xi+mx,x∈I,where Pmif;xi,xi+1,…,xi+mx, i=0,1,…,N, is the Abel–Goncharov interpolation polynomials defined in (47) and xN+j=xN−m+j−1,j=1,2,…,m.Theorem 6.
The operatorℒmAGf reproduces all univariate polynomials of degree ≤m.Proof.
ℒmAGp=p is proved from the following result:(53)∑i=0Nψix=1,Pmip;xi,xi+1,…,xi+m=p,p∈ℙm,for i=0,1,…,N.


## 3.2. The Convergence Rate of the Operator

In order to give the convergence rate of the operatorℒmAG, we apply the following notations:(54)Iρx=x−ρ,x+ρ,ρ>0,r=infρ>0:∀x∈I,Iρx∩X≠∅,M=maxx∈I#Irx∩X,for X=x0,x1,…,xN, where #⋅ denotes the cardinality function. Thus, we can obtain 2r=maxx1−x0,x2−x1,…,xN−xN−1,andM denotes the maximum number of points from X contained in an interval Irx. Then, we provide the following error estimates.Theorem 7.
Suppose thatc satisfies(55)c≤Drl,where D is a positive constant and l is a positive integer. If fx∈CmI, then(56)ℒmAGfx−fx∞≤KMfm∞El,mr,where(57)El,mr=rlnr,m=1,l=1,r,m=1,l>1,rm,m>1,m≤2l−l,r2l−1,m>1,m>2l−l,and K is a positive constant independent of x and X.Proof.
Letxi,xi+1,…,xi+m∈I be fixed, i=0,1,…,N, m∈ℕ,xi<xi+1,…,xi+m; for each x∈I, we set(58)dxi,xi+1,…,xi+m=xi+m−1−x,x≤xi,xi+m−1−xi,xi≤x≤xi+m−1,x−xi,xi+m−1≤x,dmxi,xi+1,…,xi+m=dxi,xi+1,…,xi+mm.
Based on (24), (47), and (52), we can give(59)LmAGfx−fx=∑i=0NψixPmif;xi,xi+1,…,xi+mx−fx≤∑i=0NψixPmif;xi,xi+1,…,xi+mx−fx≤Cmfm∞Sl,mx,where(60)Sl,mx=∑i=0Nψixdmxi,xi+1,…,xi+mx.
Assume that(61)n=xN−x02r+1,Qρu=u−ρ,u+ρ,u∈I,ρ>0,Tj=Qrx−2rj∪Qrx+2rj,j=0,…,n,where ∪j=−nnQrx+2rj denotes the covering of I with half open intervals. Thus, for every i∈0,1,…,N, there exists a unique j∈0,1,…,n, such that xi∈Tj. Then, we give the following inequalities as j≥2:(62)2j−1r≤x−xi≤2j+1r,2j−1−1r≤x−τi≤2j+1+1r,forτi∈xi−1,xi+1,dxi,xi+1,…,xi+mx≤2j+m−1+1r.
We have from the definition ofM(63)1≤#X∩T0≤M,1≤#X∩Tj≤2M,j=1,2,…,n.
Whenx0∈Tj,j=2,3,…,n, we have, after some calculations,(64)ψ0x≤12c2∫−∞x01x−t3dt+12c21x−τ02+c23/2∫x0x1x1−tx1−x0dt≤14c2x−x0−2+14c2x1−x0x−τ0−3≤14c22j−1−2r−2+22j−3−3r−2≤c2r−22j−3−2,where τ0∈x0,x1. When xN∈Tj,j=2,3,…,n, we get in an analogous manner(65)ψNx≤c2r−22j−3−2.When xii=1,…,N−1∈Tj,j=2,3,…,n, we also get(66)ψix≤12c21x−τi2+c23/2∫xi−1xi+1Bitdt≤14c2xi+1−xi−1x−τi−3≤c2r−22j−3−2,where τi∈xi−1,xi+1. Then, we have(67)Sl,mx≤∑xi∈T0,T1ψixdmxi,xi+1,…,xi+mx+∑j=2n∑xi∈Tjψixdmxi,xi+1,…,xi+mx≤∑xi∈T0,T1dmxi,xi+1,…,xi+mx+∑j=2n∑xi∈Tjψixdmxi,xi+1,…,xi+mx≤M2m−1rm+2M2m+1rm+2M∑j=2nc2r−22j−3−22j+2m−1rm≤2M2m+1rm+2m+1rm+c2rm−2∑j=2n2j−3−22j+2m−1m≤2M2m+1m2rm+D2rm+2l−2∑j=1njm−2,where the last inequality follows the form(68)2j−3≥j,j=3,4,…,n,2j+2m−1≤2m+1j,j=1,2,…,n,c≤Drl.Case 1.
m=1.(i)
Letl=1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Orlnr.(ii)
Letl>1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Or.Case 2.
m>1.(i)
Letm≤2l−1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Orm.(ii)
Letm>2l−1. Then, 2rm+D2rm+2l−2∑j=1njm−2=Or2l−1.In an analogous manner, we can obtain the desired error estimation as follows:Theorem 8.
Letc satisfy(69)c≤Drl,where D is a positive constant and l is a positive integer. If fx∈Cm+1I, then(70)ℒmAGfx−fx∞≤K′Mfm+1∞El,m′r,where(71)El,m′r=r2l−1,m+1≥2l−1,rm+1,m+1≤2l−1,and K′ is a positive constant independent of x and X.


## 4. Numerical Examples

In order to investigate the accuracy of our operators, we use the following functions on the interval0,1 [33].(72)Saddlef1=1.256+63x−12,Spheref2=64−81x−0.529−0.5,Clifff3=tanh−9x+12+0.5,Gentlef4=exp−81/16x−0.523,Steepf5=exp−81/4x−0.523,Exponentialf6=0.75exp−9x−224+0.75exp−9x+1249+0.5exp−9x−724+0.2exp−9x−42.For each functionfi,i=1,2,…,6, we will compare the numerical results of our new operator ℒmAG with the known operator ℒH2m−1 [10] as c=rl. We consider a uniform grid of 17 points for ℒ1AG and ℒH1, grid of 11 points for ℒ2AG, and grid of 8 points for ℒH3 on the interval 0,1. In order to obtain the estimation as accurate as possible, we calculated the approximated functions at the points i/101,i=1,2,…,100. Tables 1–6 present the mean and maximum errors, computed for different values of parameters l and m. These results show that the approximations of Abel–Goncharov multiquadric quasi-interpolants ℒmAG are comparable with that of the multiquadric quasi-interpolants ℒH2m−1.Table 1
Saddle.
ℒH2m−1f1ℒmAGf1l,mεmeanεmaxεmeanεmax(2, 1)0.0003980.0018750.0004760.002273(2, 2)0.0003360.0024260.0019600.009792(3, 1)0.0003950.0018730.0004730.002270(3, 2)0.0003300.0024290.0019440.009775(4, 1)0.0003950.0018730.0004730.002270(4, 2)0.0003300.0024290.0019440.009775Table 2
Sphere.
ℒH2m−1f2ℒmAGf2l,mεmeanεmaxεmeanεmax(2, 1)0.0004490.0008950.0005260.002686(2, 2)0.0000350.0001340.0003940.002127(3, 1)0.0004450.0008960.0005220.002687(3, 2)0.0000320.0001320.0003920.002128(4, 1)0.0004450.0008960.0005220.002687(4, 2)0.0000320.0001320.0003920.002128Table 3
Cliff.
ℒH2m−1f3ℒmAGf3l,mεmeanεmaxεmeanεmax(2, 1)0.0019520.0148730.0021100.029450(2, 2)0.0044440.0504050.0106990.110462(3, 1)0.0019430.0148630.0021010.029426(3, 2)0.0043420.0503770.0106040.110363(4, 1)0.0019430.0148630.0021010.029426(4, 2)0.0043420.0503770.0106040.110363Table 4
Gentle.
ℒH2m−1f4ℒmAGf4l,mεmeanεmaxεmeanεmax(2, 1)0.0005450.0016450.0006540.001975(2, 2)0.0001850.0007410.0020370.005950(3, 1)0.0005420.0016420.0006500.001975(3, 2)0.0001810.0007390.0020210.005949(4, 1)0.0005420.0016420.0006500.001975(4, 2)0.0001810.0007390.0020210.005949Table 5
Steep.
ℒH2m−1f5ℒmAGf5l,mεmeanεmaxεmeanεmax(2, 1)0.0017000.0064830.0021500.008659(2, 2)0.0019170.0090800.0122940.037863(3, 1)0.0016920.0064770.0021380.008651(3, 2)0.0018930.0091000.0122000.037811(4, 1)0.0016920.0064770.0021380.008651(4, 2)0.0018930.0091000.0122000.037811Table 6
Exponential.
ℒH2m−1f6ℒmAGf6l,mεmeanεmaxεmeanεmax(2, 1)0.0046630.0150480.0067390.026089(2, 2)0.0108390.0358080.0419080.152439(3, 1)0.0046450.0150440.0067140.026077(3, 2)0.0107310.0356890.0417340.152347(4, 1)0.0046450.0150440.0067140.026077(4, 2)0.0107310.0356890.0417330.152347


## 5. Conclusions

In this study, by combing multiquadric quasi-interpolantℒB with the Abel–Goncharov univariate operator, we construct a kind of Abel–Goncharov multiquadric quasi-interpolants ℒmAGf. Meanwhile, we have also proven that the operators ℒmAGf 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.



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