THE COMBINED SHEPARD–ABEL–GONCHAROV UNIVARIATE

. We extend the Shepard operator by combining it with the Abel-Goncharov univariate operator in order to increase the degree of exactness and to use some speciﬁc functionals. We study this combined operator and give some of its properties. We introduce the corresponding interpolation formula and study its remainder term.


INTRODUCTION
1.1.The Shepard univariate operator.Recall first some results regarding the multivariate Shepard operator for the univariate case.Let f be a real valued function defined on X ⊂ R and x i ∈ X, i = 0, . . ., N, be some distinct points.The univariate Shepard operator is defined by (1) (Sf ) (x) = , with µ ∈ R + (see, e.g., [12]).The basis functions A i may be written in barycentric form It is easy to check that The main properties of the operator S are: • The interpolation property (Sf ) (x i ) = f (x i ) , i = 0, . . ., N ; • The degree of exactness is gex (S) = 0.
The goal when extending the operator S by combining with other operators is to increase the degree of exactness and to use other sets of functionals.Let Λ := {λ i | i = 0, . . ., N } be a set of functionals and let P be the corresponding interpolation operator.We consider that Λ i ⊂ Λ are the subsets associated to the functionals λ i , i = 0, . . ., N .We have N i=0 Λ i = Λ and Λ i Λ j = ∅, excepting the case Λ i = {λ i } , i = 0, . . ., N, when Λ i Λ j = ∅, for i = j.We associate the interpolation operator P i to each subset Λ i , i = 0, . . ., N .
The operator S P defined by ( 4) is the combined operator of S and P (see, e.g., [12]).
Assume that Λ is a set of Birkhoff type functional, i.e., where where P is the interpolation operator corresponding to the set Λ B .
In the proof of this result the following relations are used: 1.2.The Abel-Goncharov univariate operator.Let n ∈ N, a, b ∈ R, a < b, and f : [a, b] → R be a function having the first n derivatives f (i) , i = 1, 2, . . ., n.Given the nodes x i ∈ [a, b], 0 ≤ i ≤ n, and the values f (i) (x i ), 0 ≤ i ≤ n, we consider the Abel-Goncharov interpolation problem of finding a polynomial P n f of degree n such that (see, e.g., [8] and [10]) (6) ( The determinant of this linear system is always nonzero and the problem ( 6) has therefore a unique solution.The Abel-Goncharov interpolation polynomial P n f can be written in the form where g k , k = 0, . . ., n are called Goncharov polynomials of degree k [9], determined by the conditions k (x) = 1.According to [8], [9] and [10], we have: Remark 4. [10].When all the nodes coincide, then the problem ( 6) is a Taylor interpolation problem and P n f takes the form Regarding the degree of exactness, we obtain the following result.
Theorem 1.The Abel-Goncharov operator P n has the degree of exactness n, i.e., dex(P n ) = n.
Proof.It is easily seen that for the test functions e i (x) = x i , x ∈ [a, b], we have (P n e i )(x) = e i (x), i = 0, . . ., n, while We obtain the following result regarding the remainder R n f of the Abel-Goncharov interpolation formula Proof.From Theorem 1 we have that gex(P n ) = n.By Peano's theorem we obtain .
For all x ∈ [a, b] we have , which, after some immediate manipulations, implies (9).

THE COMBINED SHEPARD-ABEL-GONCHAROV UNIVARIATE OPERATOR
In this section we shall assume that there exists f (i) (x i ), i = 0, . . ., N, on the set of N + 1 pairwise distinct points x i ∈ [a, b], 0 ≤ i ≤ N. Let us consider the set of linear functionals of Abel-Goncharov type: We attach to each node x i , i = 0, . . ., N , a set of nodes X i,n , n ∈ N, n ≤ N, i = 0, . . ., N, defined by (10) X i,n = {x i , x i+1 , . . ., x i+n } = {x i+υ : υ = 0, . . ., n}, i = 0, . . ., N, where x N +k+1 = x k , k = 0, . . ., n.We associate to each set of nodes X i,n , i = 0, . . ., N, the Abel-Goncharov interpolation operator, denoted P n i , i = 0, . . ., N, corresponding to the set of functionals Λ AG .The operators P n i , i = 0, . . ., N , exist and are unique because the points of the sets X i,n , i = 0, . . ., N , are pairwise distinct so the determinant of the interpolation system of the form ( 7) is always different from zero.We have (11) ( Remark 5.The set of linear functional of Abel-Goncharov type, Λ AG , is included in the set of linear functional of Birkhoff type.We notice that in case of the Abel-Goncharov interpolation we have the advantage that the determinant of the interpolation system of the form ( 7) is always different from zero, thus the interpolation polynomial always exists and is unique.
We consider the Abel-Goncharov polynomials of degree n, associated to the sets of nodes X i,n , i = 0, . . ., N , and the sets of linear functionals of Abel-Goncharov type given by (12) ( with the Goncharov polynomials given by Theorem 3. The Abel-Goncharov operators P n i , i = 0, . . ., N, have the degree of exactness n, i.e., (13) dex(P n i ) = n, i = 0, . . ., N.
The proof is obtained similarly to that of Theorem 1.
Remark 6.If we consider the sets X i,n i , i = 0, . . ., N , of the form (10) such that each of them has We denote by S AG n the Shepard operator of Abel-Goncharov type, given by where A i , i = 0, . . ., N , are given by ( 2) and P n i , i = 0, . . ., N , are given by (12).We call S AG n the combined Shepard-Abel-Goncharov operator.
Particular case.When all the nodes coincide, x 0 = . . .= x N , we obtain the Shepard operator of Taylor type: and the combined Shepard-Taylor operator given by The main properties of the Shepard-Taylor operator ST n are: , which shows the linearity of S AG n .
Theorem 6. S AG n f = f, for all f ∈ P n , where P n is the set of polynomials of degree at most n.The Shepard-Abel-Goncharov interpolation formula is where R AG n f denotes the remainder.When all the nodes coincide we have the Shepard-Taylor interpolation formula and the following result is known. where Proof.Theorem 6 implies gex(S AG n ) = n.Applying the Peano's theorem, we obtain , and finally (15).
Particular case.We consider n = 1.We have the corresponding Shepard-Abel-Goncharov operator given by The interpolation formula is 1 f is the remainder, which according with Theorem 8 has the following form: Example 1.Consider f : [0, 4] → R, f (x) = 3 sin πx 4 , and the nodes x i = i, i = 0, . . ., 4. Figure 1 shows the Shepard approximation corresponding to these data, while Figure 2 shows the Shepard-Abel-Goncharov approximation.The figures were drawn using Matlab.