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Summary. In this note we will present the most general linear form of a NevilleAitken-algorithm for interpolation of functions by linear combinations of functions forming a t~eby~ev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkenalgorithm [-8] applying to complete t~eby~ev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given. Subject Classifications. AMS (MOS): 65D05, 65B05, 65F05. IntroductionIn [8] the author has given an extension of the classical Neville-Aitken-algorithm [1,10] for polynomial interpolation to the more general case of interpolation by functions of a complete t~eby~ev-system which in [3] is called an outstanding problem even for simple ~eby~ev-systems. The aim of the present note is twofold: first to present a simple new proof of this extension and second to give the most general linear form of a Neville-Aitken-like interpolation algorithm computing the interpolant of a function by recurrence. This algorithm is applicable whenever a function is to be interpolated by means of linear combinations of functions forming a t~eby~ev-system such that one of its subsystems is again a (~eby~ev-system. Especially, it may be applied to interpolation by functions forming a complete or quasicomplete t~eby~ev-system where for complete (~eby~ev-systems also very general error estimates are known [9]. In case of interpolation by trigonometric polynomials the algorithm reduces to a Neville-Aitken-algorithm due to Baron [2]. Moreover, for complete t~eby~ev-systems the equivalence of the generalized Neville-Aitken-algorithm and the generalized Newton interpolation formula leads to a new simple proof of the recurrence relation for generalized divided differences [5]. Engels [4] has given another generalization of the classical Neville-Aitken-algorithm which in contrast to

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-Bsplines that reduces to the de BoorYMansionYCox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as NevilleYAitken weights of certain generalized interpolation problems. For multiple knots they are limits of NevilleYAitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval ½a; b connected at inner knots all of multiplicities zero by full connection matrices A ½i that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECTspline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves.

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