Even Degree Deficient Spline Interpolation

Abstract

In this present paper we study the general even degree spline  i.e. the spline of degree 2m, where m is the positive integer, matches derivatives upto the order of m at the knots of uniform partition. Tarazi and Sallam[6], have been constructed an interpolating quartic spline with matching first and second derivative of a given function at the knots. A similar study was made by Tarazi and Karaballi [5], for even degree splines upto degree 10. Further, it was conjectured by Tarazi and Karaballi [5], that higher degree splines can be obtained in a similar way. They also raised a question for getting a proof for general degree splines. We provide a proof for general degree spline of degree 2m. Explicit formula for these splines are obtained. Error estimation to these splines in terms of Chebyshev norm is also represented by using the result due to Cirlet, Schults and Varga [2]. On combining the result of this paper and the result obtained by Kumar and Jha [4] with some modification we get deficient spline of general degree for approximation. The deficient splines are found useful because of the fact that, in this case we require less continuity requirements (see De Boor [3], P. 125). The restrictions of smoothness are compensated by considering additional interpolatory conditions.