Generalization of (0, 4) Lacunary Interpolation by Quantic Spline

Problem statement: Spline functions are the best tool of polynomials u sed as the basic means of approximation theory in nearly all areas o f numerical analysis. Also in the problem of interpolation by g-spline construction of spline, e xistences, uniqueness and error bounds needed. Approach: In this study, we generalized (0,4) lacunary interp olation by quanta spline function. Results: The results obtained, the existence uniqueness and error bounds for generalize (0, 4) lacunary interpolation by qunatic spline. Conclusion: These generalize are preferable to interpolation by quantic spline to the use (0,4).


INTRODUCTION
Spline functions are the best tool of polynomials used as the basic means of approximation theory in nearly all areas of numerical analysis. One uses polynomial for approximation because they can be evaluated, differentiated and integrated easily and in finitely many steps using the basic arithmetic operations of addition, subtraction, division and multiplication. Spline functions constitute a relativity new subject in analysis. During the past twentieth both the theories of splines and experiences with their use in numerical analysis have under gone a considerable degree of development. The following works deal to various degree with the theory and application of splines, (Ahlberg et al., 1967). In addition to the papers mentioned above dealing with best interpolation or approximation by splines, there were also a few papers that deal with constructive properties of space of spline interpolation (Kanth et al., 2006;Khan and Aziz, 2003;Siddiqi et al., 2007). In this study we studied the generalization of one type of lacunary interpolation by quautic spline this type is (0,4) but in works (Varma, 1978;Venturino, 1996) showed this type but not in general. Also in the future we can use the same idea for different lacunary interpolation that means we can generalities for different cases in the subjected lacunary interpolation by spline.
We have Hermite interpolation if for each i, the order j of derivatives in (1) from unbroken Sequence. If some of the sequences are broken, we have lacunary interpolation.
The lacunary interpolation problem, which we have investigated in this study, consists in finding the five degree spline S(x), interpolating data given on the function value and fourth order in the interval [0,1]. Also, an extra initial condition is prescribed on the first derivative.
This study is organized as follows: First consider the spline function of degree five is presented which interpolates the lacunary data (0,4). Some theoretical results about existence, uniqueness and error bounds of the spline function of degree five are introduced and also convergence analysis is studied. To demonstrate the convergence of the prescribed lacunary spline function.

Descriptions of the method:
We present for the first time according to our knowledge a five degree spline (0,4) interpolation for one dimensional and given sufficiently smooth function f(x) defined on i = [0,1] and: p (x ) a , i 1, 2,...,n; j 0,1,2,..., n = = = We have Hermite interpolation if for each i, the order j of derivatives in (1) from unbroken Sequence. If some of the sequences are broken, we have lacunary interpolation: Denote the uniform partition of i with knots: We define the class of spline function (2) n,5 S where (2) n,5 S denotes the class of all splines of degree five which belongs to C 2 [0,1] and n is the number of knots, as follow: Any element (2) n ,5 S (x) S ∆ ∈ if the following two conditions are satisfied: Construction of the spline function: If P(x) is a polynomial of degree five on [0,1], then we have: (2 1)x 5 (2 1)x 1 A (x) (8 16 4 3)x (12 2 (2 1) 4 2)x 2 (2 1) 1 A (x) x 5 x 10 3)x 2 ( 1) (2 1) Main results: The existence and uniqueness theorem for spline function of degree five which interpolate the lacunary data (0, 4) are presented and examined.

Convergence and error bounds:
The error bound of the spline function S(x) which is a solution of the problem (6) is obtained for the uniform partition I by the following Lemma: for i = 1,2,…,n-1 Where: The result (10) follows on using the property of diagonal dominant.