The Existence , Uniqueness And Upper Bounds For Errors Of Six Degree Spline Interpolating The Lacunary Data ( 0 , 2 , 5 )

The object of this paper is to obtain the existence, uniqueness and upper bounds for errors of six degree spline interpolating the lacunary data (0,2,5). We also showed that the changes of the boundary conditions and the class of spline functions has a main role in minimizing the upper bounds for error in lacunary interpolation problem. For this reason, in the construction of our spline function which interpolates the lacunary data (0,2,5), we changed the boundary conditions and the class of spline functions which are given by [1] from first derivative to third derivative and the class of spline function from ] 1 , 0 [ 2 C to ] 1 , 0 [ 4 C .


Introduction
The subject of lacunary interpolation by polynomials has a rich history from Ltd Stone. For complete historical background we refer to the survey article [6]. The aim of this paper is to continuous works [2,5] Jwamer, but for different lacunary interpolation case that is (0,2,5) lacunary interpolation by using spline function of degree six, also we can use the same idea but for different partitions and different degree of spline functions. Many of the authors and co-workers are working on the lacunary interpolation problem by spline functions, for examples [2,4,7] and their references, but on minimizing error bounds there is a few papers, for examples [2,5].
In 2004, Jwamer [1], obtained the error bounds for (0, 2, 5) lacunary interpolation of certain classes of function by deficient six spline. In this paper we study the same lacunary interpolation data, but the essential difference here being in the boundary condition and the class of spline functions. The main object of this work is to show that the change of the boundary conditions and the class of spline functions affect on minimizing the upper bounds for error in lacunary interpolation problem.

Construction of the Spline Function
If P(x) is a polynomial of degree six on [0, 1], then we have In the subsequent section we need the following values: For fC 6 [0,1] we have the following expansions

Existence and Uniqueness:
In this section we prove the following theorem about the existence and uniqueness of the spline function Sp (6,4,n): Given arbitrary numbers there exits a unique spline Sn(x) Sp (6,4,n) such that

Error Bounds:
In this section, the upper bounds for errors studied in the following results: Proof: The result (10) follows on using the property of diagonal dominant [3].

Conclusions
In this paper, we conclude that the changes of the boundary conditions and the class of spline functions together affect on minimizing error bounds in the subject of lacunary interpolation by spline functions.