Computing ECT-B-splines recursively

Abstract

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.

Constructing totally positive piecewise Chebyshevian B-spline bases

“…The relations between the coefficients α of two consecutive intervals have already been obtained in (41), (42), and (43). With the notations introduced in (30) and (51) they can be summarised as follows:…”

“…Our overview will be as brief as possible. For further acquaintance with Extended Chebyshev spaces, see, for instance, [45,43,31,32,28,34], with Chebyshevian splines, see [45,43,5,8], with piecewise Chebyshevian splines, see [4,29,28,40,41,33,35].…”

“…Now, taking account of (36), (41), and (42), multiplication of both sides of the equality (43) by the positive number α k 1 transforms it as follows:…”

How to build all Chebyshevian spline spaces good for geometric design?

“…Via de Boor-Fix type dual functionals, he proved that the total positivity of all such connection matrices (i.e., all their minors are nonnegative), was sufficient to ensure existence of a B-spline basis and of a de Boor-type evaluation algorithm. Later on, under the same total positivity assumption, a further proof of the existence of a B-spline basis was given by Mühlbach via generalised Chebyshevian divided differences [4,27,28]. In the meantime we had shown that, in any such spline space, existence of blossoms was equivalent to existence of a B-spline basis in the space itself and in all spline spaces deduced from it by knot insertion [18].…”

“…This very general framework has been considered by several authors (see, for instance [4,18,20,[26][27][28]), among whom the very first was Barry [1]. In his approach, the section-spaces were defined by means of given weight functions and associated differential operators, as is classical for Extended Chebyshev spaces.…”