Quintic spline interpolation on uniform meshes

Abstract

1. Introduction. In a series of papers, several authors have studied and solved the problem of lacunary interpolation by splines. The subject was of particular interest to the Hungarian school of mathematicians led by Turgn. Many of them obtained important results about existence, uniqueness, explicit representation and convergence of such interpolatory polynomials when the nodes of interpolation and the knots of the spline coincide [5,6,7,13,17,18,21,22].Others have constructed lacunary deficient splines when the interpolation data is prescribed at points midway between the knots [2,10,11,12,14,15,19,25,26]• A number of papers have also shown the importance of this type of splines in numerical quadratures or in the approximate solution of differential equations [1,3,4,8,9,16,20].The basic aim of this paper is to construct and study quintic lacunary interpolatory splines that match the given values of a function f E C 5 and its second or first derivative at the equidistant knots and whose fourth derivatives interpolate f(4) midway between the knots. Such spline functions may be used to obtain approximate solutions of non-linear differential equations of the fourth order with boundary conditions that arise in the theory of bending and twisting of elastic beams and plates (see, for instance, [23,24]).