BOUNDS FOR THE REMAINDER IN THE BIVARIATE SHEPARD INTERPOLATION OF LIDSTONE

. We study the bivariate Shepard-Lidstone interpolation operator and obtain new estimates for the remainder. Some numerical examples are provided. MSC 2000. 41A63, 41A80.


INTRODUCTION
As it is pointed out in [7] and [15], interpolation at nodes having no exploitable pattern is referred to as the case of scattered data and there are two important methods of interpolation in this case: the method of Shepard and the interpolation by radial basis functions.x g = ∂ r g/∂x r and D r y g = ∂ r g/∂y r .According to [1] and [2], for a fixed ∆ denote the set According to [2], for where Λ k is the Lidstone polynomial of degree 2k + 1, k ∈ N on the interval [0, 1].
We have the interpolation formula where R ∆ m f denotes the remainder.For a fixed rectangular partition [1] and [2]).
According to [2], for ), the Lidstone interpolant L ρ m f uniquely exists and can be explicitly expressed as where r m,i,j , 0 With the previous assumptions we denote by L ∆,i m f the restriction of the Lidstone interpolation polynomial L ∆ m f to the subinterval [x i , x i+1 ], 0 ≤ i ≤ N, given by (1), and in analogous way we obtain the expression of L ∆ ,i m f, the restriction of We denote by S L the univariate combined Shepard-Lidstone operator, introduced by us in [4]: with A i , i = 0, ..., N , given by ( 5) and ( 6) The univariate Shepard-Lidstone interpolation formula is ) and the set of Lidstone functionals m f the restriction of the polynomial given by (2) to the subrectangle [x i , x i+1 ]×[y i , y i+1 ], 0 ≤ i ≤ N.This 2m−1 polynomial, in each variable, solves the interpolation problem corresponding to the set Λ i Li , 0 ≤ i ≤ N and it uniquely exists.
We have The bivariate Shepard operator of Lidstone type S Li , introduced by us in [5], is given by ( 8) We obtain the bivariate Shepard-Lidstone interpolation formula, ( 9) where S Li f is given by ( 8) and R Li f denotes the remainder of the interpolation formula.
Next, we give an error estimation using the modulus of smoothness of order k.For a function g defined on [a, b] we have We consider the norm in the set C(X) of continuous functions defined on X by We recall first a result from [17]: Theorem 5. [17] Let L be a bounded operator and let L(P ) = P for every We apply this result for the operators 2m .Now we give an estimation of the remainder R L f from (7), in terms of the modulus of smoothness.
Proof.We have and taking into account (10) and that The next result provides an estimation of the error in formula (9).and, finally, Applying Theorem 6 three times the conclusion follows.