# A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property

Authors: Ruifeng Wu; Huilai Li; Tieru Wu
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the
shifts of the functionf(x)(x∈ℝ) to approximate the derivatives of f(x), we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a nonnegative integer m. Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.’s quasi-interpolation scheme.


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## Body

## 1. Introduction

LetX={x0,…,xn} be a set of n+1 distinct points of ℝ and let f be a function defined on a domain [a,b] containing X. The standard formula for interpolating the function f, where
(1)a=x0<⋯<xn=b,
has the following form:
(2)ℒ[f;a,b](x)=∑j=0n‍λj𝒳(x-xj),
s.t.                        (3)ℒ[f;a,b](xj)=f(xj),
for all j=0,…,n, where 𝒳(·) is an interpolation kernel. Many investigators use radial basis functions to solve the interpolation problems (2)-(3). In particular, the multiquadrics presented by Hardy [1],
(4)ϕj(x)=ϕ(x-xj)=(x-xj)2+c2,j=0,…,n,
are of especial interest because of their special convergence property, see [2, 3]. Throughout this paper, let the notations ϕj(·) and c denote the multiquadrics and their shape-preserving parameter as in (4), respectively. A review by Franke [4] indicated that the multiquadric interpolation is one of the best schemes among some 29 interpolation methods in terms of accuracy, efficiency, and easy implementation. Although the multiquadric interpolation is always solvable when the scattered points {xj}j=0n are distinct [5], the resulting matrix in (2)-(3) quickly becomes ill-conditioned as the number of the scattered points increases. In this paper, we will use the quasi-interpolation technique to overcome the ill-conditioning problem.A weaker form of (3), well-known as quasi-interpolation, holds for all polynomials of degree no more than m, where m is nonnegative integer; that is,
(5)ℒ[p;a,b](xj)=p(xj),j=0,…,n,p∈ℙm,
where ℙm={p:deg⁡(p)⩽m}. Beatson and Powell [6] first proposed a univariate quasi-interpolation operator ℒB which reproduces constants, where 𝒳(x) in (2) is a linear combination of the multiquadrics, see (46)-(47). Afterwards, Wu and Schaback [7] proposed another quasi-interpolation operator ℒD which possesses shape-preserving and linear reproducing properties. They showed that the error of the operator ℒD is 𝒪(h2|ln⁡h|), when the shape parameter c=𝒪(h), where h=max⁡1⩽j⩽n⁡{|xj-xj-1|}. Using the operator ℒD, Ling [8] constructed a multilevel quasi-interpolation operator and proved that its convergence order is 𝒪(h2.5|ln⁡h|) when c=𝒪(h). Using the shifts of the cubic multiquadrics, Feng and Li [9] constructed a shape-preserving quasi-interpolation operator. They showed that the operator reproduces all polynomials of degree 2 or less and proved that the convergence rate is 𝒪(h3) as c=𝒪(h1.5). Combining the operator ℒB with Hermite interpolation polynomials, Wang et al. [10] proposed a kind of improved quasi-interpolation operators ℒH2m-1 which reproduce all polynomials of degree ⩽2m-1. They proved that it converges with a rate of 𝒪(h2m) at most. However, the operators ℒH2m-1 require values of the derivatives at endpoints, which are not convenient for practical purposes. Further, many authors offered some examples using multiquadric quasi-interpolation operator to solve differential equations, see [11–15] for details.Based on CAIRA-DELL’ACCIO’s idea [16], we first define a family of even order Bernoulli-type multiquadric quasi-interpolants ℒ~vm by combining the multiquadric quasi-interpolation operator ℒB in [6] with the polynomial expansion in even order Bernoulli polynomials vm(x) in [17]. For practical purposes, applying the divided difference formula in [18] to the operators ℒ~vm, we construct a family of modified even order Bernoulli-type multiquadric quasi-interpolants ℒvm which do not require values of the derivatives at nodes. We prove that the operators ℒvm reproduce all polynomials of degree ⩽2m and have the convergence rate of 𝒪(h2m+1) under a suitable assumption on the shape parameter c. Therefore, our operators ℒvm can provide the desired smoothness and precision in the practical applications.The organization of the remainder of this paper is as follows. In Section2, we briefly recall the definition of Bernoulli polynomials and even order Bernoulli polynomials giving some useful properties. We also obtain three useful theorems for the error in the even order Bernoulli polynomials expansion. In Section 3, we apply previous results to derive a family of modified even order Bernoulli-type multiquadric quasi-interpolants and get their convergence rate. In Section 4, numerical examples are shown to compare the approximation capacity of our new operators with that of CAIRA-DELL’ACCIO’s interpolants and Wang et al.’s quasi-interpolants. In Section 5, we apply our operators to the fitting of discrete solutions of initial value problems for ordinary differential equations. In Section 6, we give the conclusions.


## 2. Some Remarks on the Polynomial Expansion

### 2.1. The Generalized Taylor Polynomial

The generalized Taylor polynomial is an expansion in Bernoulli polynomialsBn(x), that is, the polynomials of the sequence defined recursively by means of the following relations, see [19]:
(6)B0(x)=1,Bn′(x)=nBn-1(x),n⩾1,∫01‍Bn(x)dx=0,n⩾1.
Let function f(x) be in the class Cm[a,b](a<b); then
(7)f(x)=Pm[f;a,b](x)+Rm[f;a,b](x),x∈[a,b],
where the polynomial approximation term is considered as follows:
(8)Pm[f;a,b](x)=f(a)+∑k=1m‍Bk((x-a)/h)-Bk(0)k!mmmm×hk-1(f(k-1)(b)-f(k-1)(a))
and the remainder term is defined by
(9)Rm[f;a,b](x)=hm-1m!∫ab‍f(m)(t)mm×(Bm(b-th)-Bm(x-t-[x-t]h))dt,
where [·] denotes the integer part of the argument and h=b-a. The polynomial approximant Pm[f;a,b](x) has the following result:
(10)lim⁡h→0Pm[f;a,b](x)=Tm[f;a](x),
where Tm[f;a](x) is the rth Taylor polynomial of f with initial point in a. According to (10), we denote by Pm[f;a,b](x) the generalized Taylor polynomial.


### 2.2.  The Polynomial Expansion in Even Order Bernoulli Polynomials

Let us consider the polynomial sequence defined recursively by the following relations, see [17]:
(11)v0(x)=1,vn′(x)=∫0x‍vn-1(t)dt,n=1,2,…,∫01‍vn(x)dx=0,n=1,2,….
By (11), the polynomial sequence {vn(x)} is related to the following Bernoulli polynomials of even degree, see [17]:
(12)vn(x)=22n(2n)!B2n(1+x2),n=1,2,….
We denote by vn(x) the even order Bernoulli polynomials. For any function f in the class C2m[a,b](a<b), this expansion is realized by the following:
(13)f(x)=Pa,m[f;a,b](x)+Ra,m[f;a,b](x),x∈[a,b],
where the polynomial expansion Pa,m[f;a,b](x) in even order Bernoulli polynomials is defined by
(14)Pa,m[f;a,b](x)=f(a)+∑j=1m‍[(vj(b-xh)-vj(1))h2j-1f(2j-1)(b)(vj(x-ah)-vj(0))mmim-h2j-1f(2j-1)(a)(vj(b-xh)-vj(1))(vj(x-ah)-vj(0))]
and the remainder Ra,m[f;a,b](x) in its Peano's representation is given by
(15)Ra,m[f;a,b](x)=h2m-1∫ab‍f(2m)(t)Ka,m(x-ah,t-ah)dt,
where
(16)Ka,m(x,t)={-∑j=1m‍(vj(x)-vj(0))(1-t)2m-2j(2m-2j)!,x⩽t,t2m-1(2m-1)!+∑j=1m‍(vj(1-x)-vj(1))mmmmmmmm×t2m-2j(2m-2j)!,x⩾t.
In order to get bounds for remainder (15) even in points outside the interval [a,b], we consider the operator
(17)f⟶Pa,m[f;a,b],
where f∈Cm[c,d] with c<a and b<d. By applying Peano's kernel theorem [20], we give an integral expression for the remainder (15) as follows.Theorem 1.
Iff∈C2m[c,d] and x∈[c,d], then for the remainder
(18)Ra,m[f;a,b](x)=f(x)-Pa,m[f;a,b](x)
we have the following integral representations:
(19)Ra,m[f;a,b](x)={1(2m-1)!∫xbf(2m)(t)Ka,b(x,t)dt,c⩽x⩽a,1(2m-1)!∫abf(2m)(t)Ka,b(x,t)dt,a⩽x⩽b,1(2m-1)!∫axf(2m)(t)Ka,b(x,t)dt,b⩽x⩽d,
where
(20)Ka,b(x,t)=(x-t)+2m-1-(a-t)+2m-1-∑j=1m‍[(2m-1)!(2m-2j)!h2j-1(2m-1)!(2m-2j)!(b-t)+2m-2j(vj(x-ah-vj(0)))mmm-h2j-1(2m-1)!(2m-2j)!(a-t)+2m-2jmmm×(vj(b-xh)-vj(1))(2m-1)!(2m-2j)!]
and (·)+k denotes the positive part of the kth power of the argument; that is,
(21)(s)+k=max⁡{sk,0}.Proof.
On one hand, there are evaluations of derivatives off up to the order 2m-1 on points a and b of [c,d] in the approximation term (14); on the other hand, the exactness of (14) on the set ℙ2m denotes the exactness of the operator Pa,m[f;a,b] on the subset ℙ2m-1. Applying Peano's kernel theorem, we then obtain
(22)Ra,m[f;a,b](x)=1(2m-1)!∫cd‍f(2m)(t)Ka,b(x,t)dt,
where (20) is given by applying the linear functional f→Ra,m[f;a,b](x) to a function (x-t)+2m-1 in x. Let x∈[c,a]; then
(23)Ra,m[f;a,b](x)=1(2m-1)!∫cx‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫xa‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫ab‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫bd‍f(2m)(t)Ka,b(x,t)dt.
Let t∈[c,x]; then
(24)Ka,b(x,t)=(x-t)2m-1-(a-t)2m-1-[∑j=1m‍(2m-1)!(2m-2j)!∑j=1m‍(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1(vj(x-ah)-vj(0))mm+(2m-1)!(2m-2j)!(a-t)2m-2j(vj(b-xh-vj(1)))∑j=1m‍(2m-1)!(2m-2j)!]=0,
where (x-t)2m-1 is considered a polynomial in x of degree 2m-1. By the expression of (·)+k, (20) is equal to zero in the interval b<t<d. Thus, we prove the first case of (19). The remaining cases of (19) can be got in an analogous manner.By Theorem1, we can get the following result.Theorem 2.
Iff∈C2m[c,d] and x∈[c,d], then for the remainder (18) we get
(25)|Ra,m[f;a,b](x)|⩽{C(m)∥f(2m)∥∞(b-x)2m,c<x<a,C(m)∥f(2m)∥∞(b-a)2m,a<x<b,C(m)∥f(2m)∥∞(x-a)2m,b<x<d,
where ∥·∥∞ denotes the sup-norm on [c,d] and
(26)C(m)=1(2m)![∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)(2jk)22m-1+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1mmmmmmmmm×(2m2j)(2jk)|Bk(1)|∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1].Proof.
Letc<x<a; then we find from (19) that
(27)Ra,m[f;a,b](x)=1(2m-1)!∫xαf(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫abf(2m)(t)Ka,b(x,t)dt.
Let x<t<a; then
(28)Ka,b(x,t)=-(a-t)2m-1-∑j=1m[(2m-1)!(2m-2j)!(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1mmmmmmmmmmm×(vj(x-ah)-vj(0))mmmmmmmmmmm-(2m-1)!(2m-2j)!(a-t)2m-2jh2j-1mmmmmmmmmmm×(vj(b-xh)-vj(1))(2m-1)!(2m-2j)!]
so that
(29)∫xaKa,b(x,t)f(2m)(t)dt=-∫xa(a-t)2m-1f(2m)(t)dt-∑j=1m(2m-1)!(2m-2j)!h2j-1×∫xa[(vj(x-ah)-vj(0))(vj(b-xh)-vj(1))(b-t)2m-2j(vj(x-ah)-vj(0))mmm-(a-t)2m-2j(vj(b-xh)-vj(1))(vj(x-ah)-vj(0))(vj(x-ah)-vj(0))]f(2m)(t)dt.
In [21], we have the following known identity:
(30)Bn(z+w)=∑i=0n‍(ni)Bi(z)wn-i,n=0,1,….
By the identities [17], using relations (30), we get
(31)vn(x)=22n(2n)!B2n(1+x2)=22n(2n)!B2n(12+x2)=22n(2n)!∑i=02n‍(2ni)Bi(12)(x2)2n-i,n=0,1,….
In [17], we have
(32)B0(1)=B0(12)=1,B1(0)=-12,B1(1)=12,Bn(12)=-(1-21-n)Bn(0),n=1,2,…,Bn(0)=Bn(1),n=2,…,B2n+1(0)=0,n=1,2,….
Therefore, by applying (31), we obtain the following form from (29):
(33)vj(x-ah)-vj(0)=22j(2j)!∑k=02j-1‍(2jk)Bk(12)(x-a2h)2j-k,j=1,2,…,vj(b-xh)-vj(1)=22j(2j)!∑k=02j-1‍(2jk)Bk(1)(a-x2h)2j-k,j=1,2,….
Further, by applying the third case of (32) and the identities (33), we have
(34)∫xa‍Ka,b(x,t)f(2m)(t)dt=-∫xa‍(a-t)2m-1f(2m)(t)dt-∑j=1m(2m-1)!(2m-2j)!22j(2j)!(x-a2h)2jh2j-1m×∫xa‍((b-t)2m-2j-(a-t)2m-2j)f(2m)(t)dt-∑j=1m∑k=12j-1(2m-1)!(2m-2j)!22j(2j)!Bk(12)(x-a2h)2j-kh2j-1mm×∫xa‍(b-t)2m-2jf(2m)(t)dt+∑j=1m∑k=12j-1(2m-1)!(2m-2j)!22j(2j)!Bk(1)(a-x2h)2j-kh2j-1mm×∫xa‍(a-t)2m-2jf(2m)(t)dt.
Note that the integrands are of type g(t)f(2m)(t) with a g(t) that does not change sign in [x,a]. By applying the first mean value theorem for integrals to (34), we find for some ξ, ξj, ηj, θj∈[c,d], j=1,…,m that
(35)∫xa‍Ka,b(x,t)f(2m)(t)dt=-f(2m)(ξ)∫xa‍(a-t)2m-1dt-∑j=1m‍(2m-1)!(2m-2j)!22j(2j)!(x-a2h)2jh2j-1f(2m)(ξj)m×∫xa‍((b-t)2m-2j-(a-t)2m-2j)dt-∑j=1m‍∑k=12j-1‍(2m-1)!(2m-2j)!22j(2j)!Bk(12)(x-a2h)2j-kmm×h2j-1f(2m)(ηj)∫xa‍(b-t)2m-2jdt+∑j=1m‍∑k=12j-1‍(2m-1)!(2m-2j)!22j(2j)!Bk(1)(a-x2h)2j-kmm×h2j-1f(2m)(θj)∫xa‍(a-t)2m-2jdt.
After some calculations in (35), we obtain
(36)∫xa‍Ka,b(x,t)f(2m)(t)dt=-f(2m)(ξ)(a-x)2m2m-h2m∑j=1m‍(2m-1)!(2m-2j+1)!(2j)!(x-ah)2jf(2m)(ξj)mmmm×[-1+∑r=02m-2j‍(b-xh)2m-2j-r(a-xh)r]-h2m∑j=1m‍∑k=12j-1‍(2m-1)!2k(2m-2j)!(2j!)(2jk)Bk(12)(x-ah)2j-kmmmm×f(2m)(ηj)[1-(b-xh)2m-2j+1]+h2m∑j=1m‍∑k=12j-1‍(2m-1)!2k(2m-2j+1)!(2j)!(2jk)Bk(1)mmmm×(a-xh)2j-kf(2m)(θj)(a-xh)2m-2j+1.
Let a<t<b; then we have
(37)Ka,b(x,t)=-∑j=1m‍(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1mmmmm×(vj(x-ah)-vj(0)),∫ab‍Ka,b(x,t)f(2m)(t)dt=-∑j=1m‍∑k=02j-1‍(2m-1)!(2m-2j)!h2j-1(2jk)Bk(12)mm×(x-ah)2j-k∫xa‍(b-t)2m-2jf(2m)(t)dt.
By the first mean value theorem for integrals, we can get after some calculations
(38)∫ab‍Ka,b(x,t)f(2m)(t)dt=-h2m∑j=1m‍∑k=02j-1‍(2m-1)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmmm×(x-ah)2j-kf(2m)(λj),
where λj∈[c,d], j=1,…,m. By applying relations (36) and (38) to (27), we have
(39)Ra,m[f;a,b](x)=-f(2m)(ξ)(a-x)2m(2m)!-h2m(2m)!∑j=1m‍(2m)!(2m-2j+1)!(2j)!(x-ah)2jf(2m)(ξj)mmmmm×[-1+∑r=02m-2j‍(b-xh)2m-2j+r(a-xh)r]-h2m(2m)!∑j=1m‍∑k=12j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmmmm×(x-ah)2j-kf(2m)(ηj)mmmmm×[1-(b-xh)2m-2j+1]+h2m(2m)!∑j=1m‍∑k=12j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(1)mmmim×(a-xh)2m-k+1f(2m)(θj)-h2m(2m)!×∑j=1m‍∑k=02j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmm×(x-ah)2j-kf(2m)(λj).
By identities (32), we obtain
(40)|B0(12)|=|B0(1)|=1,|Bn(12)|=2n-22n|Bn(1)|,n=1,2,….
By using (40) in (39), we have after some simplifications that
(41)|Ra,m[f;a,b](x)|⩽∥f(2m)∥∞(2m)!(∑j=1m‍(2m2j)(b-xh)2m(a-x)2m+h2m∑j=1m‍(2m2j)(b-xh)2mmmmmmm+h2m∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)(2jk)|Bk(1)|mmmmmm×(b-xh)2m+1-k∑j=1m‍(2m2j)(b-xh)2m)⩽h2m∥f(2m)∥∞(2m)!(∑j=1m‍(2m2j)(b-xh)2m1+∑j=1m‍(2m2j)+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)mmmmmmmm×(2jk)|Bk(1)|∑j=1m‍(2m2j)(b-xh)2m)(b-xh)2m.
Because
(42)1+∑j=1m‍(2m2j)=22m-1,
we obtain the first case of expression (25). Similarly, we can prove the remaining cases.Since the polynomialsPa,m[f;a,b](x) of degree are not greater than 2m, we can obtain the desired bounds in an analogous manner.Theorem 3.
Iff∈C2m+1[c,d] and x∈[c,d], then for the remainder (18) we get
(43)|Ra,m[f;a,b](x)|⩽{C′(m)∥f(2m+1)∥∞(b-x)2m+1,c<x<a,C′(m)∥f(2m+1)∥∞(b-a)2m+1,a<x<b,C′(m)∥f(2m+1)∥∞(x-a)2m+1,b<x<d,
where ∥·∥∞ denotes the sup-norm on [c,d] and
(44)C′(m)=1(2m+1)![∑aaa22m+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+2(2m+12j)mmmmmmmmmmmmm×(2jk)|Bk(1)|∑aaa].


## 2.1. The Generalized Taylor Polynomial

The generalized Taylor polynomial is an expansion in Bernoulli polynomialsBn(x), that is, the polynomials of the sequence defined recursively by means of the following relations, see [19]:
(6)B0(x)=1,Bn′(x)=nBn-1(x),n⩾1,∫01‍Bn(x)dx=0,n⩾1.
Let function f(x) be in the class Cm[a,b](a<b); then
(7)f(x)=Pm[f;a,b](x)+Rm[f;a,b](x),x∈[a,b],
where the polynomial approximation term is considered as follows:
(8)Pm[f;a,b](x)=f(a)+∑k=1m‍Bk((x-a)/h)-Bk(0)k!mmmm×hk-1(f(k-1)(b)-f(k-1)(a))
and the remainder term is defined by
(9)Rm[f;a,b](x)=hm-1m!∫ab‍f(m)(t)mm×(Bm(b-th)-Bm(x-t-[x-t]h))dt,
where [·] denotes the integer part of the argument and h=b-a. The polynomial approximant Pm[f;a,b](x) has the following result:
(10)lim⁡h→0Pm[f;a,b](x)=Tm[f;a](x),
where Tm[f;a](x) is the rth Taylor polynomial of f with initial point in a. According to (10), we denote by Pm[f;a,b](x) the generalized Taylor polynomial.


## 2.2.  The Polynomial Expansion in Even Order Bernoulli Polynomials

Let us consider the polynomial sequence defined recursively by the following relations, see [17]:
(11)v0(x)=1,vn′(x)=∫0x‍vn-1(t)dt,n=1,2,…,∫01‍vn(x)dx=0,n=1,2,….
By (11), the polynomial sequence {vn(x)} is related to the following Bernoulli polynomials of even degree, see [17]:
(12)vn(x)=22n(2n)!B2n(1+x2),n=1,2,….
We denote by vn(x) the even order Bernoulli polynomials. For any function f in the class C2m[a,b](a<b), this expansion is realized by the following:
(13)f(x)=Pa,m[f;a,b](x)+Ra,m[f;a,b](x),x∈[a,b],
where the polynomial expansion Pa,m[f;a,b](x) in even order Bernoulli polynomials is defined by
(14)Pa,m[f;a,b](x)=f(a)+∑j=1m‍[(vj(b-xh)-vj(1))h2j-1f(2j-1)(b)(vj(x-ah)-vj(0))mmim-h2j-1f(2j-1)(a)(vj(b-xh)-vj(1))(vj(x-ah)-vj(0))]
and the remainder Ra,m[f;a,b](x) in its Peano's representation is given by
(15)Ra,m[f;a,b](x)=h2m-1∫ab‍f(2m)(t)Ka,m(x-ah,t-ah)dt,
where
(16)Ka,m(x,t)={-∑j=1m‍(vj(x)-vj(0))(1-t)2m-2j(2m-2j)!,x⩽t,t2m-1(2m-1)!+∑j=1m‍(vj(1-x)-vj(1))mmmmmmmm×t2m-2j(2m-2j)!,x⩾t.
In order to get bounds for remainder (15) even in points outside the interval [a,b], we consider the operator
(17)f⟶Pa,m[f;a,b],
where f∈Cm[c,d] with c<a and b<d. By applying Peano's kernel theorem [20], we give an integral expression for the remainder (15) as follows.Theorem 1.
Iff∈C2m[c,d] and x∈[c,d], then for the remainder
(18)Ra,m[f;a,b](x)=f(x)-Pa,m[f;a,b](x)
we have the following integral representations:
(19)Ra,m[f;a,b](x)={1(2m-1)!∫xbf(2m)(t)Ka,b(x,t)dt,c⩽x⩽a,1(2m-1)!∫abf(2m)(t)Ka,b(x,t)dt,a⩽x⩽b,1(2m-1)!∫axf(2m)(t)Ka,b(x,t)dt,b⩽x⩽d,
where
(20)Ka,b(x,t)=(x-t)+2m-1-(a-t)+2m-1-∑j=1m‍[(2m-1)!(2m-2j)!h2j-1(2m-1)!(2m-2j)!(b-t)+2m-2j(vj(x-ah-vj(0)))mmm-h2j-1(2m-1)!(2m-2j)!(a-t)+2m-2jmmm×(vj(b-xh)-vj(1))(2m-1)!(2m-2j)!]
and (·)+k denotes the positive part of the kth power of the argument; that is,
(21)(s)+k=max⁡{sk,0}.Proof.
On one hand, there are evaluations of derivatives off up to the order 2m-1 on points a and b of [c,d] in the approximation term (14); on the other hand, the exactness of (14) on the set ℙ2m denotes the exactness of the operator Pa,m[f;a,b] on the subset ℙ2m-1. Applying Peano's kernel theorem, we then obtain
(22)Ra,m[f;a,b](x)=1(2m-1)!∫cd‍f(2m)(t)Ka,b(x,t)dt,
where (20) is given by applying the linear functional f→Ra,m[f;a,b](x) to a function (x-t)+2m-1 in x. Let x∈[c,a]; then
(23)Ra,m[f;a,b](x)=1(2m-1)!∫cx‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫xa‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫ab‍f(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫bd‍f(2m)(t)Ka,b(x,t)dt.
Let t∈[c,x]; then
(24)Ka,b(x,t)=(x-t)2m-1-(a-t)2m-1-[∑j=1m‍(2m-1)!(2m-2j)!∑j=1m‍(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1(vj(x-ah)-vj(0))mm+(2m-1)!(2m-2j)!(a-t)2m-2j(vj(b-xh-vj(1)))∑j=1m‍(2m-1)!(2m-2j)!]=0,
where (x-t)2m-1 is considered a polynomial in x of degree 2m-1. By the expression of (·)+k, (20) is equal to zero in the interval b<t<d. Thus, we prove the first case of (19). The remaining cases of (19) can be got in an analogous manner.By Theorem1, we can get the following result.Theorem 2.
Iff∈C2m[c,d] and x∈[c,d], then for the remainder (18) we get
(25)|Ra,m[f;a,b](x)|⩽{C(m)∥f(2m)∥∞(b-x)2m,c<x<a,C(m)∥f(2m)∥∞(b-a)2m,a<x<b,C(m)∥f(2m)∥∞(x-a)2m,b<x<d,
where ∥·∥∞ denotes the sup-norm on [c,d] and
(26)C(m)=1(2m)![∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)(2jk)22m-1+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1mmmmmmmmm×(2m2j)(2jk)|Bk(1)|∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1].Proof.
Letc<x<a; then we find from (19) that
(27)Ra,m[f;a,b](x)=1(2m-1)!∫xαf(2m)(t)Ka,b(x,t)dt+1(2m-1)!∫abf(2m)(t)Ka,b(x,t)dt.
Let x<t<a; then
(28)Ka,b(x,t)=-(a-t)2m-1-∑j=1m[(2m-1)!(2m-2j)!(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1mmmmmmmmmmm×(vj(x-ah)-vj(0))mmmmmmmmmmm-(2m-1)!(2m-2j)!(a-t)2m-2jh2j-1mmmmmmmmmmm×(vj(b-xh)-vj(1))(2m-1)!(2m-2j)!]
so that
(29)∫xaKa,b(x,t)f(2m)(t)dt=-∫xa(a-t)2m-1f(2m)(t)dt-∑j=1m(2m-1)!(2m-2j)!h2j-1×∫xa[(vj(x-ah)-vj(0))(vj(b-xh)-vj(1))(b-t)2m-2j(vj(x-ah)-vj(0))mmm-(a-t)2m-2j(vj(b-xh)-vj(1))(vj(x-ah)-vj(0))(vj(x-ah)-vj(0))]f(2m)(t)dt.
In [21], we have the following known identity:
(30)Bn(z+w)=∑i=0n‍(ni)Bi(z)wn-i,n=0,1,….
By the identities [17], using relations (30), we get
(31)vn(x)=22n(2n)!B2n(1+x2)=22n(2n)!B2n(12+x2)=22n(2n)!∑i=02n‍(2ni)Bi(12)(x2)2n-i,n=0,1,….
In [17], we have
(32)B0(1)=B0(12)=1,B1(0)=-12,B1(1)=12,Bn(12)=-(1-21-n)Bn(0),n=1,2,…,Bn(0)=Bn(1),n=2,…,B2n+1(0)=0,n=1,2,….
Therefore, by applying (31), we obtain the following form from (29):
(33)vj(x-ah)-vj(0)=22j(2j)!∑k=02j-1‍(2jk)Bk(12)(x-a2h)2j-k,j=1,2,…,vj(b-xh)-vj(1)=22j(2j)!∑k=02j-1‍(2jk)Bk(1)(a-x2h)2j-k,j=1,2,….
Further, by applying the third case of (32) and the identities (33), we have
(34)∫xa‍Ka,b(x,t)f(2m)(t)dt=-∫xa‍(a-t)2m-1f(2m)(t)dt-∑j=1m(2m-1)!(2m-2j)!22j(2j)!(x-a2h)2jh2j-1m×∫xa‍((b-t)2m-2j-(a-t)2m-2j)f(2m)(t)dt-∑j=1m∑k=12j-1(2m-1)!(2m-2j)!22j(2j)!Bk(12)(x-a2h)2j-kh2j-1mm×∫xa‍(b-t)2m-2jf(2m)(t)dt+∑j=1m∑k=12j-1(2m-1)!(2m-2j)!22j(2j)!Bk(1)(a-x2h)2j-kh2j-1mm×∫xa‍(a-t)2m-2jf(2m)(t)dt.
Note that the integrands are of type g(t)f(2m)(t) with a g(t) that does not change sign in [x,a]. By applying the first mean value theorem for integrals to (34), we find for some ξ, ξj, ηj, θj∈[c,d], j=1,…,m that
(35)∫xa‍Ka,b(x,t)f(2m)(t)dt=-f(2m)(ξ)∫xa‍(a-t)2m-1dt-∑j=1m‍(2m-1)!(2m-2j)!22j(2j)!(x-a2h)2jh2j-1f(2m)(ξj)m×∫xa‍((b-t)2m-2j-(a-t)2m-2j)dt-∑j=1m‍∑k=12j-1‍(2m-1)!(2m-2j)!22j(2j)!Bk(12)(x-a2h)2j-kmm×h2j-1f(2m)(ηj)∫xa‍(b-t)2m-2jdt+∑j=1m‍∑k=12j-1‍(2m-1)!(2m-2j)!22j(2j)!Bk(1)(a-x2h)2j-kmm×h2j-1f(2m)(θj)∫xa‍(a-t)2m-2jdt.
After some calculations in (35), we obtain
(36)∫xa‍Ka,b(x,t)f(2m)(t)dt=-f(2m)(ξ)(a-x)2m2m-h2m∑j=1m‍(2m-1)!(2m-2j+1)!(2j)!(x-ah)2jf(2m)(ξj)mmmm×[-1+∑r=02m-2j‍(b-xh)2m-2j-r(a-xh)r]-h2m∑j=1m‍∑k=12j-1‍(2m-1)!2k(2m-2j)!(2j!)(2jk)Bk(12)(x-ah)2j-kmmmm×f(2m)(ηj)[1-(b-xh)2m-2j+1]+h2m∑j=1m‍∑k=12j-1‍(2m-1)!2k(2m-2j+1)!(2j)!(2jk)Bk(1)mmmm×(a-xh)2j-kf(2m)(θj)(a-xh)2m-2j+1.
Let a<t<b; then we have
(37)Ka,b(x,t)=-∑j=1m‍(2m-1)!(2m-2j)!(b-t)2m-2jh2j-1mmmmm×(vj(x-ah)-vj(0)),∫ab‍Ka,b(x,t)f(2m)(t)dt=-∑j=1m‍∑k=02j-1‍(2m-1)!(2m-2j)!h2j-1(2jk)Bk(12)mm×(x-ah)2j-k∫xa‍(b-t)2m-2jf(2m)(t)dt.
By the first mean value theorem for integrals, we can get after some calculations
(38)∫ab‍Ka,b(x,t)f(2m)(t)dt=-h2m∑j=1m‍∑k=02j-1‍(2m-1)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmmm×(x-ah)2j-kf(2m)(λj),
where λj∈[c,d], j=1,…,m. By applying relations (36) and (38) to (27), we have
(39)Ra,m[f;a,b](x)=-f(2m)(ξ)(a-x)2m(2m)!-h2m(2m)!∑j=1m‍(2m)!(2m-2j+1)!(2j)!(x-ah)2jf(2m)(ξj)mmmmm×[-1+∑r=02m-2j‍(b-xh)2m-2j+r(a-xh)r]-h2m(2m)!∑j=1m‍∑k=12j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmmmm×(x-ah)2j-kf(2m)(ηj)mmmmm×[1-(b-xh)2m-2j+1]+h2m(2m)!∑j=1m‍∑k=12j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(1)mmmim×(a-xh)2m-k+1f(2m)(θj)-h2m(2m)!×∑j=1m‍∑k=02j-1‍(2m)!2k(2m-2j+1)!(2j)!(2jk)Bk(12)mmm×(x-ah)2j-kf(2m)(λj).
By identities (32), we obtain
(40)|B0(12)|=|B0(1)|=1,|Bn(12)|=2n-22n|Bn(1)|,n=1,2,….
By using (40) in (39), we have after some simplifications that
(41)|Ra,m[f;a,b](x)|⩽∥f(2m)∥∞(2m)!(∑j=1m‍(2m2j)(b-xh)2m(a-x)2m+h2m∑j=1m‍(2m2j)(b-xh)2mmmmmmm+h2m∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)(2jk)|Bk(1)|mmmmmm×(b-xh)2m+1-k∑j=1m‍(2m2j)(b-xh)2m)⩽h2m∥f(2m)∥∞(2m)!(∑j=1m‍(2m2j)(b-xh)2m1+∑j=1m‍(2m2j)+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+1(2m2j)mmmmmmmm×(2jk)|Bk(1)|∑j=1m‍(2m2j)(b-xh)2m)(b-xh)2m.
Because
(42)1+∑j=1m‍(2m2j)=22m-1,
we obtain the first case of expression (25). Similarly, we can prove the remaining cases.Since the polynomialsPa,m[f;a,b](x) of degree are not greater than 2m, we can obtain the desired bounds in an analogous manner.Theorem 3.
Iff∈C2m+1[c,d] and x∈[c,d], then for the remainder (18) we get
(43)|Ra,m[f;a,b](x)|⩽{C′(m)∥f(2m+1)∥∞(b-x)2m+1,c<x<a,C′(m)∥f(2m+1)∥∞(b-a)2m+1,a<x<b,C′(m)∥f(2m+1)∥∞(x-a)2m+1,b<x<d,
where ∥·∥∞ denotes the sup-norm on [c,d] and
(44)C′(m)=1(2m+1)![∑aaa22m+∑j=1m‍∑k=12j-1‍2k+1-22m-2j+2(2m+12j)mmmmmmmmmmmmm×(2jk)|Bk(1)|∑aaa].


## 3. The Modified Even Order Bernoulli-Type Quasi-Interpolants

The multiquadric quasi-interpolantℒB [6] is defined by the following:
(45)ℒB[f;a,b](x)=f(x0)ψ0(x)+∑i=1n-1f(xi)ψi(x)+f(xn)ψn(x),x∈[a,b],
where
(46)ψ0(x)=12c2∫-∞x0‍1[(x-t)2+c2]3/2dt+12c2∫x0x1‍(x1-t)/(x1-x0)[(x-t)2+c2]3/2dt=12+ϕ1(x)-ϕ0(x)2(x1-x0),(47)ψn(x)=12c2∫xn∞‍1[(x-t)2+c2]3/2+12c2∫xn-1xn‍(t-xn-1)/(xn-xn-1)[(x-t)2+c2]3/2dt=12-ϕn(x)-ϕn-1(x)2(xn-xn-1),ψi(x)=12c2∫xi-1xi+1‍Bi(t)[(x-t)2+c2]3/2dt=ϕi+1(x)-ϕi(x)2(xi+1-xi)-ϕi(x)-ϕi-1(x)2(xi-xi-1)
for i=1,…,n-1, where {Bi(t):t∈ℝ} is the hat function that has the nodes {xi-1,xi,xi+1}, that is identically zero outside the interval xi-1⩽t⩽xi+1 and that satisfies the normalization condition Bi(xi)=1. The operator ℒB reproduces constants. Based on the operator ℒB, we first define a family of even order Bernoulli-type multiquadric quasi-interpolants ℒ~vm as follows:
(48)ℒ~vm[f;a,b](x)=∑i=0n‍ψi(x)Pxi,m[f;xi,xi+1](x),mmmmmmmmmmmx∈[a,b],
where Pxi,m[f;xi,xi+1](x) is the natural extension of the polynomial expansion defined in (14) and xn+1=xn-1. The operators ℒ~vm possess the polynomial reproduction property as follows.Theorem 4.
The operatorsℒ~vm reproduce all univariate polynomials of degree no more than 2m.Proof.
The argumentℒ~vm[p;a,b](x)=p follows from the well-known property
(49)∑i=0n‍ψi(x)=1
since Pxi,m[p;xi,xi+1](x)=p for i=0,…,n, where p∈ℙ2m.Although the quasi-interpolantsℒ~vm reproduce all polynomials of degree ⩽2m, they require the derivative of f at every node, which are very difficult to measure in practice. Therefore, we use divided difference operator DA2j-1f in following Definition 5 to approximate f(2j-1) in the operators ℒ~vm and then get a family of modified even order Bernoulli-type multiquadric quasi-interpolants ℒvm.Definition 5 (see [18]).
Letℱ={f∣f:ℝ→ℝ} and let A be a discrete subset of ℝ, j∈ℕ. Suppose that D2j-1 is the order 2j-1 derivative. An operator DA2j-1:ℱ→ℱ is said to be a ℙ2m-exact A-discretization of D2j-1 if and only if(i)
there exists a real vectorλ=(λa)a∈A s.t. for any f∈ℱ,
(50)DA2j-1f(·)=∑a∈A‍λaf(·+a),j=1,…,m;(ii)
for anyp∈ℙ2m,
(51)DA2j-1p=D2j-1p.
In such situation, we also say thatDA2j-1f is a ℙ2m-exact A-discretization of D2j-1f. Let the points be distinct in the set A; then DA2j-1 is determined uniquely.
Let|·| denote the number of elements in set. Let the points in set A be distinct and |A|=2m+1; then by Definition 5 and [18], a ℙ2m-exact A-discretization of the order 2j-1 derivative f(2j-1) is
(52)DA2j-1f(x)=∑a∈A‍λaf(x+a),j=1,…,m,
where
(53)λa=(-1)2m+1-2j(2j-1)!∑A′⊂A∖{a},|A′|=2m+1-2j‍∏c∈A′‍(c)∏c∈A∖{a}‍(a-c).
By virtue of the location of each pairxi, xi+1(i=0,…,n), we choose suitable sets Axi,Axi+1and then replace f(2j-1)(xi) and f(2j-1)(xi+1) with DAxi2j-1f(xi) and DAxi+12j-1f(xi+1), respectively. Thus, the modification quasi-interpolants ℒvm can be expressed as follows:
(54)ℒvm[f;a,b](x)=∑i=0n‍ψi(x)m×[∑aaaf(xi)+∑j=1m‍[∑aaa(xi+1-xi)2j-1DAxi+12j-1f(xi+1)22j(2j)!mmmmmmmm×∑k=02j-1‍(2jk)Bk(12)(x-xi2(xi+1-xi))2j-kmmmmmmmm-(xi+1-xi)2j-1DAxi2j-1f(xi)22j(2j)!mmmmmmmm×∑k=02j-1‍(2jk)Bk(1)(xi-x2(xi+1-xi))2j-k∑aaa]∑aaa],mmmmmmmmmmmmmmmmmmimiimmmmxn+1=xn-1.
Note that the expressions of DAxi2j-1f(xi)(i=0,…,n,j=1,…,m) in the modification operators ℒvm are provided by the following theorem.Theorem 6.
For anyA⊂ℝ and x∈ℝ, let A be a ℙ2m-unisolvent set and Ax=A-x denote the set of points e∈ℝ of the form e=a-x, where a∈A. Let A={xi,…,xi+2m}(i=0,…,m-1), {xi-m,…,xi,…,xi+m}(i=m,…,n-m), and {xi-2m,…,xi}(i=n-m+1,…,n). Then, for each j=1,…,m, we have
(55)DAxi2j-1f(xi)=∑a∈A‍λa-xif(a),i=0,…,n,
where
(56)λa-xi=(-1)2m+1-2j(2j-1)!∑A′⊂A∖{a},|A′|=2m+1-2j∏c∈A′‍(c-xi)∏c∈A∖{a}‍(a-c).Proof.
For eachx∈ℝ, we set Ax={xi-x,…,xi+2m-x}(i=0,…,m-1), {xi-m-x,…,xi-x,…,xi+m-x}(i=m,⋯,n-m), and {xi-2m-x,…,xi-x}(i=n-m+1,…,n). According to (50) and (53), we get
(57)DAx2j-1f(0)=∑a∈A‍λa-xf(a-x),
where
(58)λa-x=(-1)2m+1-2j(2j-1)!∑A′⊂A∖{a},|A′|=2m+1-2j‍∏c∈A′‍(c-x)∏c∈A∖{a}‍(a-c).
Therefore, we have
(59)DAx2j-1f(x)=∑a∈A‍λa-xf(a).
Let us set x=xi; then we get the proof of the Theorem 6.Remark 7.
Form=1, we give the expression of the modification operator ℒv1 as follows:
(60)ℒv1[f;a,b](x)=[f(x0)-(12(x0-x)2x1-x0+(x0-x))D31f(x0)]ψ0(x)+∑i=1n-1‍[f(xi)-(12(xi-x)2xi+1-xi+(xi-x))D31f(xi)]ψi(x)+∑i=1n-1‍12(x-xi-1)2xi-xi-1D31f(xi)ψi-1(x)+12(x-xn-1)2xn-xn-1D31f(xn)ψn-1(x)+[f(xn)-(12(xn-x)2xn-1-xn+(xn-x))D31f(xn)]ψn(x)+12(x-xn)2xn-1-xnD31f(xn-1)ψn(x),
where
(61)D31f(x0)=-x1+x2-2x0(x0-x1)(x0-x2)f(x0)-x2-x0(x1-x0)(x1-x2)f(x1)-x1-x0(x2-x0)(x2-x1)f(x2),D31f(xi)=-xi+1-xi(xi-1-xi)(xi-1-xi+1)f(xi-1)-xi-1+xi+1-2xi(xi-xi-1)(xi-xi+1)f(xi)-xi-1-xi(xi+1-xi-1)(xi+1-xi-1)(xi+1-xi)f(xi+1),mmmmmmmmmmmmmmmmmmi=1,…,n-1,D31f(xn)=-xn-1-xn(xn-2-xn-1)(xn-2-xn)f(xn-2)-xn-2-xn(xn-1-xn-2)(xn-1-xn)f(xn-1)-xn-2+xn-1-2xn(xn-xn-2)(xn-xn-1)f(xn).


### 3.1. The Polynomial Reproduction Properties of the Operatorsℒvm

Theorem 8.
The operatorsℒvm reproduce all univariate polynomials of degree no more than 2m.Proof.
By using the proof of Theorem4 and formulas (51)–(54), we get the proof of Theorem 8 immediately.


### 3.2.  The Convergence Rate of the Operatorsℒvm

In order to obtain the convergence rate of the modified multiquadric quasi-interpolantsℒvm, we make use of the following notations:
(62)Iρ(x)=[x-ρ,x+ρ],ρ>0,h=12inf⁡{ρ>0:∀x∈[a,b],Iρ(x)∩X≠∅},M=max⁡x∈[a,b]#(Ih(x)∩X),
where #(·) denotes the cardinality function. So, 2h=max⁡1⩽i⩽n⁡|xi-xi-1| and M denotes the maximum number of points from X contained in an interval Ih(x). At first, for the quasi-interpolants ℒ~vm, we then give the error estimates as follows.Theorem 9.
Letc satisfy
(63)c⩽Drl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m[a,b]; then
(64)∥ℒ~vm[f;a,b](x)-f(x)∥∞⩽C′M∥f(2m)(x)∥∞Tl,m(h),
where
(65)Tl,m(h)={h2m,2m<2l-1,h2l-1,2m⩾2l-1,
and C′ is a positive constant independent of f, x, and X.Proof.
Let each pairxi, xi+1∈[a,b], be fixed and let xi<xi+1. For each x∈[a,b] we make use of the following settings:
(66)d[xi,xi+1](x)={xi+1-x,x⩽xi,xi+1-xi,xi⩽x⩽xi+1,x-xi,xi+1⩽x,d2m[xi,xi+1](x)=(d[xi,xi+1](x))2m.
By applying (14) to (48), we obtain
(67)|ℒ~vm[f;a,b](x)-f(x)|=|∑i=0n‍ψi(x)(Pxi,m[f;xi,xi+1])-f(x)|⩽∑i=0n‍ψi(x)|Pxi,m[f;xi,xi+1](x)-f(x)|⩽C(m)∥f(2m)(x)∥∞Sl,m(x),
where
(68)Sl,m(x)=∑i=0n‍ψi(x)d2m[xi,xi+1](x).
Assume that
(69)N=[b-a2h]+1,Qρ(u)=(u-ρ,u+ρ],u∈[a,b],ρ>0,Tj=Qh(x-2hj)∪Qh(x+2hj),j=0,…,N,
where the set ⋃j=-NNQh(x+2hj) denotes the covering of [a,b] with half open intervals. Thus, for every i∈{0,…,n}, there exists a unique j∈{0,…,N} s.t. xi∈Tj. Then, we get the following inequalities:
(70)(2j-1)h⩽|x-xi|⩽(2j+1)h,(2(j-1)-1)h⩽|x-τi|⩽(2(j+1)+1)h,
where j=2,…,N and τi∈[xi-1,xi+1]. Therefore, we have from (70)
(71)d[xi,xi+1](x)⩽(2j+3)h.
We also obtain from the definition of M(72)1⩽#(X∩T0)⩽M,1⩽#(X∩Tj)⩽2M,j=1,…,N.
On the other hand, when x0∈Tj,j=2,…,N, we get, after some calculations, by applying the first mean value theorem for integrals to (46),
(73)ψ0(x)⩽12c2∫-∞x0‍1|x-t|3dt+12c21[(x-τ0)2+c2]3/2×∫x0x1‍x1-tx1-x0dt⩽14c2|x-x0|-2+14c2(x1-x0)|x-τ0|-3⩽14c2[(2j-1)-2h-2+2(2j-3)-3h-2]⩽c2h-2(2j-3)-2,
where τ0∈[x0,x1]. When xn∈Tj,j=2,…,N, we obtain in an analogous manner
(74)ψn(x)⩽c2h-2(2j-3)-2.
When xi(i=1,…,n-1)∈yj(j=2,…,N), we also obtain
(75)ψi(x)⩽12c21[(x-τi)2+c2]3/2∫xi-1xi+1‍Bi(t)dt⩽14c2(xi+1-xi-1)|x-τi|-3⩽c2h-2(2j-3)-2,
where τi∈[xi-1,xi+1]. Then, for (68), we have
(76)Sl,m(x)=∑xi∈T0,T1‍ψi(x)d2m[xi,xi+1](x)+∑j=2N‍∑xi∈Tj‍ψi(x)d2m[xi,xi+1](x)⩽∑xi∈T0,T1‍d2m[xi,xi+1](x)+∑j=2N‍∑xi∈Tj‍ψi(x)d2m[xi,xi+1](x)⩽M(3h)2m+2M(5h)2m+2M∑j=2N‍c2h-2(2j-3)-2((2j+3)h)2m⩽2M(5h)2m+2M(5h)2m+2M∑j=1N‍c2h-2j-2((5j)h)2m⩽2M52m(2h2m+D2h2m-2+2l∑j=1N‍j2m-2),
where the last inequality follows from
(77)2j-3⩾j,j=3,4,…,2j+3⩽5j,j=1,2,…,72m⩽52m+122102m,m=1,2,….
Let 2m<2l-1; then 2h2m+D2h2l+2m-2∑j=1N‍j2m-2=𝒪(h2m).
Let2m⩾2l-1; then 2h2m+D2h2l+2m-2∑j=1N‍j2m-2=𝒪(h2l-1).Applying Theorem3, we can obtain the desired error estimates of the operator ℒ~vm in an analogous manner.Theorem 10.
Letc satisfy
(78)c⩽Dhl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m+1[a,b]; then
(79)∥ℒ~vm[f;a,b](x)-f(x)∥∞⩽C′′M∥f(2m+1)(x)∥∞Tl,m′(h),
where
(80)Tl,m′(h)={h2m+1,2m+1<2l-1,h2l-1,2m+1⩾2l-1,
and C′′ is a positive constant independent of f, x, and X.Because of disadvantage with the derivatives in the operatorsℒ~vm, we give the following desired error estimates of the modification quasi-interpolants ℒvm.Theorem 11.
Letc satisfy
(81)c⩽Dhl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m+1[a,b]; then
(82)∥ℒvm[f;a,b](x)-f(x)∥∞⩽CM∥f(2m+1)(x)∥∞Tl,m′(h),
where
(83)Tl,m′(h)={h2m+1,2m+1<2l-1,h2l-1,2m+1⩾2l-1,
and C is a positive constant independent of f, x, and X.Proof.
Consider(84)|ℒvm[f;a,b](x)-f(x)|⩽|ℒvm[f;a,b](x)-ℒ~vm[f;a,b](x)|+|ℒ~vm[f;a,b](x)-f(x)|.
The second term of the right-hand sides in (84) has been obtained from Theorem 10, so we only need to prove the first term.
We denote byhmax⁡ and hmin⁡ the maximum and the minimum distance between adjacent nodes, respectively. Let h=hmax⁡/hmin⁡⩾1 and C0=∥f(2m+1)(x)∥∞.
LetC1 be a constant; then, according to [18], we get
(85)∑a∈A‍|λa||a|2m+1(2m+1)!⩽∑a∈A‍|λa|1(2m+1)!(max⁡a∈A|A|)2m+1⩽C1h2m+1-(2j-1)k2m.
Therefore, we obtain
(86)|DA2j-1f(xi)-f(2j-1)(xi)|⩽C1h2m+1-(2j-1)k2m×∥f(2m+1)(x)∥∞.
Let C2,C3,…,C2m,C¯,C_,C be constants; then
(87)|ℒvm[f;a,b](x)-ℒ~vm[f;a,b](x)|⩽C0C2h2mk2m∑i=0n‍|x-xi|ψi(x)+C0C3h2m-1k2mmmmmm×∑i=0n‍|x-xi|2ψi(x)+⋯+C0C2mh2k2mmmmmm×∑i=0n‍|x-xi|2m-1ψi(x)⩽C0C2h2mk2m×(∑xi∈T0,T1‍|x-xi|ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|ψi(x))+C0C3h2m-1k2m×(∑xi∈T0,T1‍|x-xi|2ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|2ψi(x))+⋯+C0C2mh2k2m×(∑xi∈T0,T1‍|x-xi|2m-1ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|2m-1ψi(x))⩽C0C2h2mk2m×(Mh+2M(3h)+2M∑j=2N‍c2h-2(2j-3)-2(2j+1)h)+C0C3h2m-1k2m×(Mh2+2M(3h)2+∑j=2N‍c2h-2(2j-3)-2((2j+1)h)2)+⋯+C0C2mh2k2m×(∑j=2N‍Mh2m-1+2M(3h)2m-1mm+2M∑j=2N‍c2h-2(2j-3)-2((2j+1)h)2m-1∑j=2N‍)⩽C¯M∥f(2m+1)(x)∥∞(h2m+1+c2h2m-2+c2h2m-3+⋯+c2h)⩽C_M∥f(2m+1)(x)∥∞(h2m+1+c2h)⩽C_M∥f(2m+1)(x)∥∞(h2m+1+D2h2l+1).
Finally, applying Theorem 10 and (87) to (84), we get
(88)|ℒvm[f;a,b](x)-f(x)|⩽CM∥f(2m+1)(x)∥∞×(2h2m+1+D2h2l+1+D2h2l-1).
Let 2m+1<2l-1, then 2h2m+1+D2h2l-1=𝒪(h2m+1).
Let2m+1⩾2l-1, then 2h2m+1+D2h2l-1=𝒪(h2l-1).


## 3.1. The Polynomial Reproduction Properties of the Operatorsℒvm

Theorem 8.
The operatorsℒvm reproduce all univariate polynomials of degree no more than 2m.Proof.
By using the proof of Theorem4 and formulas (51)–(54), we get the proof of Theorem 8 immediately.


## 3.2.  The Convergence Rate of the Operatorsℒvm

In order to obtain the convergence rate of the modified multiquadric quasi-interpolantsℒvm, we make use of the following notations:
(62)Iρ(x)=[x-ρ,x+ρ],ρ>0,h=12inf⁡{ρ>0:∀x∈[a,b],Iρ(x)∩X≠∅},M=max⁡x∈[a,b]#(Ih(x)∩X),
where #(·) denotes the cardinality function. So, 2h=max⁡1⩽i⩽n⁡|xi-xi-1| and M denotes the maximum number of points from X contained in an interval Ih(x). At first, for the quasi-interpolants ℒ~vm, we then give the error estimates as follows.Theorem 9.
Letc satisfy
(63)c⩽Drl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m[a,b]; then
(64)∥ℒ~vm[f;a,b](x)-f(x)∥∞⩽C′M∥f(2m)(x)∥∞Tl,m(h),
where
(65)Tl,m(h)={h2m,2m<2l-1,h2l-1,2m⩾2l-1,
and C′ is a positive constant independent of f, x, and X.Proof.
Let each pairxi, xi+1∈[a,b], be fixed and let xi<xi+1. For each x∈[a,b] we make use of the following settings:
(66)d[xi,xi+1](x)={xi+1-x,x⩽xi,xi+1-xi,xi⩽x⩽xi+1,x-xi,xi+1⩽x,d2m[xi,xi+1](x)=(d[xi,xi+1](x))2m.
By applying (14) to (48), we obtain
(67)|ℒ~vm[f;a,b](x)-f(x)|=|∑i=0n‍ψi(x)(Pxi,m[f;xi,xi+1])-f(x)|⩽∑i=0n‍ψi(x)|Pxi,m[f;xi,xi+1](x)-f(x)|⩽C(m)∥f(2m)(x)∥∞Sl,m(x),
where
(68)Sl,m(x)=∑i=0n‍ψi(x)d2m[xi,xi+1](x).
Assume that
(69)N=[b-a2h]+1,Qρ(u)=(u-ρ,u+ρ],u∈[a,b],ρ>0,Tj=Qh(x-2hj)∪Qh(x+2hj),j=0,…,N,
where the set ⋃j=-NNQh(x+2hj) denotes the covering of [a,b] with half open intervals. Thus, for every i∈{0,…,n}, there exists a unique j∈{0,…,N} s.t. xi∈Tj. Then, we get the following inequalities:
(70)(2j-1)h⩽|x-xi|⩽(2j+1)h,(2(j-1)-1)h⩽|x-τi|⩽(2(j+1)+1)h,
where j=2,…,N and τi∈[xi-1,xi+1]. Therefore, we have from (70)
(71)d[xi,xi+1](x)⩽(2j+3)h.
We also obtain from the definition of M(72)1⩽#(X∩T0)⩽M,1⩽#(X∩Tj)⩽2M,j=1,…,N.
On the other hand, when x0∈Tj,j=2,…,N, we get, after some calculations, by applying the first mean value theorem for integrals to (46),
(73)ψ0(x)⩽12c2∫-∞x0‍1|x-t|3dt+12c21[(x-τ0)2+c2]3/2×∫x0x1‍x1-tx1-x0dt⩽14c2|x-x0|-2+14c2(x1-x0)|x-τ0|-3⩽14c2[(2j-1)-2h-2+2(2j-3)-3h-2]⩽c2h-2(2j-3)-2,
where τ0∈[x0,x1]. When xn∈Tj,j=2,…,N, we obtain in an analogous manner
(74)ψn(x)⩽c2h-2(2j-3)-2.
When xi(i=1,…,n-1)∈yj(j=2,…,N), we also obtain
(75)ψi(x)⩽12c21[(x-τi)2+c2]3/2∫xi-1xi+1‍Bi(t)dt⩽14c2(xi+1-xi-1)|x-τi|-3⩽c2h-2(2j-3)-2,
where τi∈[xi-1,xi+1]. Then, for (68), we have
(76)Sl,m(x)=∑xi∈T0,T1‍ψi(x)d2m[xi,xi+1](x)+∑j=2N‍∑xi∈Tj‍ψi(x)d2m[xi,xi+1](x)⩽∑xi∈T0,T1‍d2m[xi,xi+1](x)+∑j=2N‍∑xi∈Tj‍ψi(x)d2m[xi,xi+1](x)⩽M(3h)2m+2M(5h)2m+2M∑j=2N‍c2h-2(2j-3)-2((2j+3)h)2m⩽2M(5h)2m+2M(5h)2m+2M∑j=1N‍c2h-2j-2((5j)h)2m⩽2M52m(2h2m+D2h2m-2+2l∑j=1N‍j2m-2),
where the last inequality follows from
(77)2j-3⩾j,j=3,4,…,2j+3⩽5j,j=1,2,…,72m⩽52m+122102m,m=1,2,….
Let 2m<2l-1; then 2h2m+D2h2l+2m-2∑j=1N‍j2m-2=𝒪(h2m).
Let2m⩾2l-1; then 2h2m+D2h2l+2m-2∑j=1N‍j2m-2=𝒪(h2l-1).Applying Theorem3, we can obtain the desired error estimates of the operator ℒ~vm in an analogous manner.Theorem 10.
Letc satisfy
(78)c⩽Dhl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m+1[a,b]; then
(79)∥ℒ~vm[f;a,b](x)-f(x)∥∞⩽C′′M∥f(2m+1)(x)∥∞Tl,m′(h),
where
(80)Tl,m′(h)={h2m+1,2m+1<2l-1,h2l-1,2m+1⩾2l-1,
and C′′ is a positive constant independent of f, x, and X.Because of disadvantage with the derivatives in the operatorsℒ~vm, we give the following desired error estimates of the modification quasi-interpolants ℒvm.Theorem 11.
Letc satisfy
(81)c⩽Dhl,
where D is a positive constant and l is a positive integer. Let f(x)∈C2m+1[a,b]; then
(82)∥ℒvm[f;a,b](x)-f(x)∥∞⩽CM∥f(2m+1)(x)∥∞Tl,m′(h),
where
(83)Tl,m′(h)={h2m+1,2m+1<2l-1,h2l-1,2m+1⩾2l-1,
and C is a positive constant independent of f, x, and X.Proof.
Consider(84)|ℒvm[f;a,b](x)-f(x)|⩽|ℒvm[f;a,b](x)-ℒ~vm[f;a,b](x)|+|ℒ~vm[f;a,b](x)-f(x)|.
The second term of the right-hand sides in (84) has been obtained from Theorem 10, so we only need to prove the first term.
We denote byhmax⁡ and hmin⁡ the maximum and the minimum distance between adjacent nodes, respectively. Let h=hmax⁡/hmin⁡⩾1 and C0=∥f(2m+1)(x)∥∞.
LetC1 be a constant; then, according to [18], we get
(85)∑a∈A‍|λa||a|2m+1(2m+1)!⩽∑a∈A‍|λa|1(2m+1)!(max⁡a∈A|A|)2m+1⩽C1h2m+1-(2j-1)k2m.
Therefore, we obtain
(86)|DA2j-1f(xi)-f(2j-1)(xi)|⩽C1h2m+1-(2j-1)k2m×∥f(2m+1)(x)∥∞.
Let C2,C3,…,C2m,C¯,C_,C be constants; then
(87)|ℒvm[f;a,b](x)-ℒ~vm[f;a,b](x)|⩽C0C2h2mk2m∑i=0n‍|x-xi|ψi(x)+C0C3h2m-1k2mmmmmm×∑i=0n‍|x-xi|2ψi(x)+⋯+C0C2mh2k2mmmmmm×∑i=0n‍|x-xi|2m-1ψi(x)⩽C0C2h2mk2m×(∑xi∈T0,T1‍|x-xi|ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|ψi(x))+C0C3h2m-1k2m×(∑xi∈T0,T1‍|x-xi|2ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|2ψi(x))+⋯+C0C2mh2k2m×(∑xi∈T0,T1‍|x-xi|2m-1ψi(x)+∑j=2N‍∑xi∈Tj‍|x-xi|2m-1ψi(x))⩽C0C2h2mk2m×(Mh+2M(3h)+2M∑j=2N‍c2h-2(2j-3)-2(2j+1)h)+C0C3h2m-1k2m×(Mh2+2M(3h)2+∑j=2N‍c2h-2(2j-3)-2((2j+1)h)2)+⋯+C0C2mh2k2m×(∑j=2N‍Mh2m-1+2M(3h)2m-1mm+2M∑j=2N‍c2h-2(2j-3)-2((2j+1)h)2m-1∑j=2N‍)⩽C¯M∥f(2m+1)(x)∥∞(h2m+1+c2h2m-2+c2h2m-3+⋯+c2h)⩽C_M∥f(2m+1)(x)∥∞(h2m+1+c2h)⩽C_M∥f(2m+1)(x)∥∞(h2m+1+D2h2l+1).
Finally, applying Theorem 10 and (87) to (84), we get
(88)|ℒvm[f;a,b](x)-f(x)|⩽CM∥f(2m+1)(x)∥∞×(2h2m+1+D2h2l+1+D2h2l-1).
Let 2m+1<2l-1, then 2h2m+1+D2h2l-1=𝒪(h2m+1).
Let2m+1⩾2l-1, then 2h2m+1+D2h2l-1=𝒪(h2l-1).


## 4. Numerical Examples

We consider the following functions on the interval[0,1], which are firstly used in [16]:
(89)Saddlef1=1.256+6(3x-1)2,Spheref2=64-81(x-0.5)29-0.5.
We apply the interpolation operators SBm, the quasi-interpolation operators ℒH2m-1, and the quasi-interpolants ℒvm on the above functions with c=(2h)l, where SBm and ℒH2m-1 are defined by [10, 16], respectively.We use uniform grids of 21 points for the operatorsSBm, ℒH2m-1, and ℒvm in Tables 1, 2, 3, and 4. In order to estimate the errors as accurate as possible, we compute the approximation functions at the points i/101, i=1,…,100. Tables 1–4 show the mean and max errors which are computed for different values of the parameters μ, l, and m. The numerical results show that our quasi-interpolants ℒvm have better approximation behavior.Table 1
Numerical results of the operatorsSBm and ℒvm for the saddle function.
S
B
m
f
1
ℒ
v
m
f
1
(
μ
,
m
)
ε
mean
ε
max
⁡
(
l
,
m
)
ε
mean
ε
max
⁡
(
2,1
)
1.002
×
1
0
-
3
4.786
×
1
0
-
3
(
2,1
)
2.237
×
1
0
-
4
1.126
×
1
0
-
3
(
2,2
)
5.836
×
1
0
-
4
3.654
×
1
0
-
3
(
2,2
)
3.356
×
1
0
-
6
2.745
×
1
0
-
5
(
2,3
)
4.148
×
1
0
-
4
2.776
×
1
0
-
3
(
2,3
)
4.020
×
1
0
-
7
2.300
×
1
0
-
6
(
3,1
)
4.100
×
1
0
-
4
2.578
×
1
0
-
3
(
3,1
)
2.063
×
1
0
-
4
1.097
×
1
0
-
3
(
3,2
)
1.626
×
1
0
-
4
1.200
×
1
0
-
3
(
3,2
)
1.698
×
1
0
-
6
2.623
×
1
0
-
5
(
3,3
)
6.893
×
1
0
-
5
2.332
×
1
0
-
4
(
3,3
)
1.325
×
1
0
-
8
2.021
×
1
0
-
7
(
4,1
)
3.008
×
1
0
-
4
3.643
×
1
0
-
3
(
4,1
)
1.824
×
1
0
-
4
1.053
×
1
0
-
3
(
4,2
)
1.822
×
1
0
-
4
2.004
×
1
0
-
4
(
4,2
)
9.620
×
1
0
-
7
2.363
×
1
0
-
5
(
4,3
)
1.844
×
1
0
-
5
5.491
×
1
0
-
4
(
4,3
)
9.891
×
1
0
-
8
2.031
×
1
0
-
7Table 2
Numerical results of the operatorsℒH2m-1 and ℒvm for the saddle function.
ℒ
H
2
m
-
1
f
1
ℒ
v
m
f
1
(
l
,
m
)
ε
mean
ε
max
⁡
(
l
,
m
)
ε
mean
ε
max
⁡
(
2,1
)
2.654
×
1
0
-
4
1.200
×
1
0
-
3
(
2,1
)
2.237
×
1
0
-
4
1.126
×
1
0
-
3
(
2,2
)
6.678
×
1
0
-
6
4.005
×
1
0
-
5
(
2,2
)
3.356
×
1
0
-
6
2.745
×
1
0
-
5
(
2,3
)
3.400
×
1
0
-
6
1.280
×
1
0
-
5
(
2,3
)
4.020
×
1
0
-
7
2.300
×
1
0
-
6
(
3,1
)
2.541
×
1
0
-
4
1.180
×
1
0
-
3
(
3,1
)
2.063
×
1
0
-
4
1.097
×
1
0
-
3
(
3,2
)
6.435
×
1
0
-
6
4.160
×
1
0
-
5
(
3,2
)
1.698
×
1
0
-
6
2.623
×
1
0
-
5
(
3,3
)
1.236
×
1
0
-
7
1.230
×
1
0
-
6
(
3,3
)
1.325
×
1
0
-
8
2.021
×
1
0
-
7
(
4,1
)
2.540
×
1
0
-
4
1.180
×
1
0
-
3
(
4,1
)
1.824
×
1
0
-
4
1.053
×
1
0
-
3
(
4,2
)
6.435
×
1
0
-
6
4.160
×
1
0
-
5
(
4,2
)
9.620
×
1
0
-
7
2.363
×
1
0
-
5
(
4,3
)
1.185
×
1
0
-
7
1.230
×
1
0
-
6
(
4,3
)
9.891
×
1
0
-
8
2.031
×
1
0
-
7Table 3
Numerical results of the operatorsSBm and ℒvm for the sphere function.
S
B
m
f
2
ℒ
v
m
f
2
(
μ
,
m
)
ε
mean
ε
max
⁡
(
l
,
m
)
ε
mean
ε
max
⁡
(
2,1
)
2.012
×
1
0
-
3
6.732
×
1
0
-
3
(
2,1
)
2.356
×
1
0
-
4
4.664
×
1
0
-
4
(
2,2
)
1.521
×
1
0
-
4
9.474
×
1
0
-
4
(
2,2
)
1.001
×
1
0
-
6
2.045
×
1
0
-
6
(
2,3
)
1.900
×
1
0
-
4
8.204
×
1
0
-
4
(
2,3
)
9.097
×
1
0
-
8
9.763
×
1
0
-
7
(
3,1
)
4.834
×
1
0
-
4
1.255
×
1
0
-
3
(
3,1
)
2.304
×
1
0
-
4
4.024
×
1
0
-
4
(
3,2
)
2.801
×
1
0
-
5
1.189
×
1
0
-
4
(
3,2
)
9.972
×
1
0
-
8
3.665
×
1
0
-
7
(
3,3
)
2.711
×
1
0
-
5
1.080
×
1
0
-
4
(
3,3
)
8.851
×
1
0
-
9
3.541
×
1
0
-
9
(
4,1
)
3.991
×
1
0
-
4
1.332
×
1
0
-
3
(
4,1
)
2.636
×
1
0
-
4
3.964
×
1
0
-
4
(
4,2
)
1.989
×
1
0
-
5
1.278
×
1
0
-
4
(
4,2
)
8.614
×
1
0
-
8
3.524
×
1
0
-
7
(
4,3
)
1.249
×
1
0
-
5
5.604
×
1
0
-
5
(
4,3
)
9.811
×
1
0
-
9
2.004
×
1
0
-
8Table 4
Numerical results of the operatorsℒH2m-1 and ℒvm for the sphere function.
ℒ
H
2
m
-
1
f
2
ℒ
v
m
f
2
(
l
,
m
)
ε
mean
ε
max
⁡
(
l
,
m
)
ε
mean
ε
max
⁡
(
2,1
)
3.037
×
1
0
-
4
5.727
×
1
0
-
4
(
2,1
)
2.356
×
1
0
-
4
4.664
×
1
0
-
4
(
2,2
)
1.590
×
1
0
-
6
4.065
×
1
0
-
6
(
2,2
)
1.001
×
1
0
-
6
2.045
×
1
0
-
6
(
2,3
)
9.838
×
1
0
-
7
2.290
×
1
0
-
6
(
2,3
)
9.097
×
1
0
-
8
9.763
×
1
0
-
7
(
3,1
)
2.848
×
1
0
-
4
5.653
×
1
0
-
4
(
3,1
)
2.304
×
1
0
-
4
4.024
×
1
0
-
4
(
3,2
)
2.217
×
1
0
-
7
3.077
×
1
0
-
6
(
3,2
)
9.972
×
1
0
-
8
3.665
×
1
0
-
7
(
3,3
)
1.581
×
1
0
-
8
3.135
×
1
0
-
8
(
3,3
)
8.851
×
1
0
-
9
3.541
×
1
0
-
9
(
4,1
)
2.848
×
1
0
-
4
5.653
×
1
0
-
4
(
4,1
)
2.636
×
1
0
-
4
3.964
×
1
0
-
4
(
4,2
)
6.017
×
1
0
-
7
3.107
×
1
0
-
6
(
4,2
)
8.614
×
1
0
-
8
3.524
×
1
0
-
7
(
4,3
)
1.730
×
1
0
-
8
6.000
×
1
0
-
8
(
4,3
)
9.811
×
1
0
-
9
2.004
×
1
0
-
8


## 5. An Application of the New Operators

After solving the following initial value problems:(90)y′(x)=f(x,y(x)),(x,y)∈[a,b]×R,y(x0)=y0,x0∈[a,b],
by virtue of a discrete method, we often need to master the solution on a set of points that differs from the grid. Here we use our operators ℒvm to solve the problems. In fact, combinations of our operators ℒvm with discrete solvers of ODEs provide approximations of the solution of the problems (90) on [a,b]. An algorithm for constructing these quasi-interpolants is given as follows. The discrete solver produces an approximation y~i of the exact solution y(xi) at nodes xi, i=0,…,n in [a,b]. Substituting the exact values mentioned above into the definition of our operators ℒvm by their respective approximations, we get the proposed quasi-interpolants. We consider the initial value problems as follows.Problem A
(91)
y
′
(
x
)
=
x
e
-
2
x
-
2
y
,
(
x
,
y
)
∈
[
0,1
]
×
ℝ
,
y
(
0
)
=
-
1
.Problem B
(92)
y
′
(
x
)
=
sin
⁡
(
2
x
)
-
y
tan
(
x
)
,
(
x
,
y
)
∈
[
0,1
]
×
ℝ
,
y
(
0
)
=
-
2
.
The exact solutions of Problems A and B are y(x)=(x-1)ex and y(x)=-2cos⁡2x, respectively. By the Runge-Kutta method of order 4, we obtain the y~i on a uniform grid of 21 nodes in [0,1]. Calculating the approximative functions at points i/101, i=1,…,100, we get the mean and max errors in Table 5. Comparing the approximation capacity of our proposed quasi-interpolants with that of Runge-Kutta scheme of order 4 and Wang et al.'s quasi-interpolation scheme [10] in Table 5, we find that our technique has smaller errors in the Problems A and B.Table 5
Numerical results of the initial value Problems A and B.
Problem A
Problem B
ε
mean
ε
max
⁡
ε
mean
ε
max
⁡
y
~
i
6.34
×
1
0
-
7
1.94
×
1
0
-
6
2.09
×
1
0
-
6
1.23
×
1
0
-
5
ℒ
H
2
m
-
1
,
(
l
,
m
)
=
(
4,2
)
2.30
×
1
0
-
7
6.18
×
1
0
-
7
7.00
×
1
0
-
7
2.43
×
1
0
-
6
ℒ
H
2
m
-
1
,
(
l
,
m
)
=
(
4,3
)
8.05
×
1
0
-
8
5.72
×
1
0
-
7
3.84
×
1
0
-
7
4.02
×
1
0
-
6
ℒ
v
m
,
(
l
,
m
)
=
(
4,2
)
6.02
×
1
0
-
8
9.17
×
1
0
-
8
9.32
×
1
0
-
8
3.43
×
1
0
-
7
ℒ
v
m
,
(
l
,
m
)
=
(
4,3
)
1.67
×
1
0
-
9
2.44
×
1
0
-
7
8.72
×
1
0
-
8
6.59
×
1
0
-
7


## 6. Conclusions

In this paper, we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolantsℒvm which reproduce polynomials of higher degree. There is no demand for the derivatives of function f approximated at each node in our operators ℒvm, so they do not increase the orders of smoothness of the function f. Under a certain assumption, we give an expected result on the convergence rate of our operators ℒvm. The numerical examples show that our operators ℒvm produce higher degree of accuracy. Furthermore, applying the operators ℒvm to the fitting of discrete solutions of initial value problems, we find that our operators ℒvm provide more accurate approximation solver.



---

*Source: 389215-2014-03-10.xml*