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

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the function f(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.

Although the multiquadric interpolation is always solvable when the scattered points {  }  =0 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 illconditioning problem.

The Polynomial Expansion in Even Order Bernoulli Polynomials.
Let us consider the polynomial sequence defined recursively by the following relations, see [17]: By (11), the polynomial sequence {V  ()} is related to the following Bernoulli polynomials of even degree, see [17]: We denote by V  () the even order Bernoulli polynomials.For any function  in the class  2 [, ] ( < ), this expansion is realized by the following: where the polynomial expansion  , [; , ]() in even order Bernoulli polynomials is defined by and the remainder  , [; , ]() in its Peano's representation is given by where In order to get bounds for remainder (15) even in points outside the interval [, ], we consider the operator where  ∈   [, ] with  <  and  < .By applying Peano's kernel theorem [20], we give an integral expression for the remainder (15) as follows.
Theorem 1.If  ∈  2 [, ] and  ∈ [, ], then for the remainder we have the following integral representations: where and (⋅)  + denotes the positive part of the th power of the argument; that is, Proof.On one hand, there are evaluations of derivatives of  up to the order 2 − 1 on points  and  of [, ] in the approximation term (14); on the other hand, the exactness of ( 14) on the set P 2 denotes the exactness of the operator  , [; , ] on the subset P 2−1 .Applying Peano's kernel theorem, we then obtain where (20) where ( − ) 2−1 is considered a polynomial in  of degree 2 − 1.By the expression of (⋅)  + , ( 20) is equal to zero in the interval  <  < .Thus, we prove the first case of (19).The remaining cases of (19) where ‖ ⋅ ‖ ∞ denotes the sup-norm on [, ] and Proof.Let  <  < ; then we find from (19) that Let  <  < ; then so that In [21], we have the following known identity: By the identities [17], using relations (30), we get 2− ,  = 0, 1, . . . .
In [17], we have Therefore, by applying (31), we obtain the following form from (29): Further, by applying the third case of (32) and the identities (33), we have Note that the integrands are of type () (2) () with a () that does not change sign in [, ].By applying the first mean value theorem for integrals to (34), we find for some ,   ,   ,   ∈ [, ],  = 1, . . .,  that After some calculations in (35), we obtain Let  <  < ; then we have By the first mean value theorem for integrals, we can get after some calculations where   ∈ [, ],  = 1, . . ., .By applying relations (36) and ( 38) to (27), we have By identities (32), we obtain Because we obtain the first case of expression (25).Similarly, we can prove the remaining cases.
Since the polynomials  , [; , ]() of degree are not greater than 2, we can obtain the desired bounds in an analogous manner.

The Modified Even Order Bernoulli-Type Quasi-Interpolants
The multiquadric quasi-interpolant L  [6] is defined by the following: where for  = 1, . . .,  − 1, where {  () :  ∈ R} is the hat function that has the nodes { −1 ,   ,  +1 }, that is identically zero outside the interval  −1 ⩽  ⩽  +1 and that satisfies the normalization condition   (  ) = 1.The operator L  reproduces constants.Based on the operator L  , we first define a family of even order Bernoulli-type multiquadric quasi-interpolants LV  as follows: where    , [;   ,  +1 ]() is the natural extension of the polynomial expansion defined in (14) and  +1 =  −1 .The operators LV  possess the polynomial reproduction property as follows.Although the quasi-interpolants LV  reproduce all polynomials of degree ⩽2, they require the derivative of  at every node, which are very difficult to measure in practice.Therefore, we use divided difference operator  2−1   in following Definition 5 to approximate  (2−1) in the operators LV  and then get a family of modified even order Bernoullitype multiquadric quasi-interpolants L V  .
Definition 5 (see [18]).Let F = { |  : R → R} and let  be a discrete subset of R,  ∈ N. Suppose that  2−1 is the order 2 − 1 derivative.An operator  2−1  : F → F is said to be a P 2 -exact -discretization of  2−1 if and only if (i) there exists a real vector  = (  ) ∈ s.t. for any  ∈ F, (ii) for any  ∈ P 2 , In such situation, we also say that where By virtue of the location of each pair   ,  +1 ( = 0, . . ., ), we choose suitable sets    ,   +1 and then replace  (2−1) (  ) and ( +1 ), respectively.Thus, the modification quasi-interpolants L V  can be expressed as follows: Note that the expressions of where Therefore, we have Let us set  =   ; then we get the proof of the Theorem 6.
Remark 7.For  = 1, we give the expression of the modification operator L V 1 as follows: where (  ) . (61)

The Polynomial Reproduction Properties of the Operators L V 𝑚
Theorem 8.The operators L V  reproduce all univariate polynomials of degree no more than 2.
Proof.By using the proof of Theorem 4 and formulas ( 51)-(54), we get the proof of Theorem 8 immediately.

The Convergence Rate of the Operators
In order to obtain the convergence rate of the modified multiquadric quasi-interpolants L V  , we make use of the following notations: where #(⋅) denotes the cardinality function. where and   is a positive constant independent of , , and .
where  is a positive constant and  is a positive integer.Let () ∈ C 2+1 [, ]; then where and   is a positive constant independent of , , and .
Because of disadvantage with the derivatives in the operators LV  , we give the following desired error estimates of the modification quasi-interpolants L V  .
where  is a positive constant and  is a positive integer.Let () ∈  2+1 [, ]; then where and  is a positive constant independent of , , and .
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.

Numerical Examples
We consider the following functions on the interval [0, 1], which are firstly used in [16]: We apply the interpolation operators    , the quasiinterpolation operators L  2−1 , and the quasi-interpolants L V  on the above functions with  = (2ℎ)  , where    and L  2−1 are defined by [10,16], respectively.We use uniform grids of 21 points for the operators    , L  2−1 , and L V  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 /101,  = 1, . . ., 100.Tables 1-4 show the mean and max errors which are computed for different values of the parameters , , and .The numerical results show that our quasi-interpolants L V  have better approximation behavior.

An Application of the New Operators
After solving the following initial value problems: 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 L V  to solve the problems.In fact, combinations of our operators L V  with discrete solvers of ODEs provide approximations of the solution of the problems (90) on [, ].An algorithm for constructing these quasiinterpolants is given as follows.The discrete solver produces an approximation ỹ of the exact solution (  ) at nodes   ,  = 0, . . .,  in [, ].Substituting the exact values mentioned above into the definition of our operators L V  by their respective approximations, we get the proposed quasi-interpolants.We consider the initial value problems as follows.(92) The exact solutions of Problems A and B are () = ( − 1)  and () = −2cos 2 , respectively.By the Runge-Kutta method of order 4, we obtain the ỹ on a uniform grid of 21 nodes in [0, 1].Calculating the approximative functions at points /101,  = 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.

Conclusions
In this paper, we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants L V  which reproduce polynomials of higher degree.There is no demand for the derivatives of function  approximated at each node in our operators L V  , so they do not increase the orders of smoothness of the function .Under a certain assumption, we give an expected result on the convergence rate of our operators L V  .The numerical examples show that our operators L V  produce higher degree of accuracy.Furthermore, applying the operators L V  to the fitting of discrete solutions of initial value problems, we find that our operators L V  provide more accurate approximation solver.
can be got in an analogous manner.
So, 2ℎ = max 1⩽⩽ |  −  −1 | and  denotes the maximum number of points from  contained in an interval  ℎ ().At first, for the quasi-interpolants LV  , we then give the error estimates as follows.

Table 1 :
Numerical results of the operators    and L V  for the saddle function.

Table 4 :
Numerical results of the operators L  2−1 and L V  for the sphere function.L  2−1  2 L V   2