Construction of B-splines for generalized spline spaces generated from local ECT-systems

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

“…Then, because the piecewise functions N , ∈ n (K), satisfy (BSB) 1 , (BSB) 3 , and (BSB) 4 , one can prove that all osculating flats Osc n−α S(t + k ), Osc n−m i S(t i ), k + 1 ≤ i ≤ k + s, and Osc n−β S(t − k+s+1 ) have in common the only point P j k −α . The arguments are exactly those used in the proofs of Proposition 2.3 and Theorem 3.1 of [16] which we refer the reader to (also see the comment after the latter theorem).…”

“…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.…”

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

“…The latter offers a much more general framework -sometimes referred to in the literature as piecewise Tchebycheffian splines [31] -and allows us to optimally benefit from the great diversity of ET-spaces, but the existence of a B-spline-like basis requires constraints on the various ET-spaces.Tchebycheffian splines can be easily incorporated in existing spline codes because the corresponding B-spline-like basis, whenever it exists, is compatible with classical B-splines as it enjoys the same structural properties. When all the ET-spaces have the same dimension, various approaches have been used in the Tchebycheffian setting to construct such a B-spline-like basis: generalized divided differences [32,34], Hermite interpolation [10,33], integral recurrence relations [3,20], de Boor-like recurrence relations [15,18], and blossoming [30]. Each of these definitions has advantages according to the problem one has to face or to the properties to be proved.…”

“…All these constructions lead to the same functions, up to a proper scaling. For non-uniform local dimensions, the literature is much less developed and B-spline-like bases have been constructed via Hermite interpolation [10,33].Unfortunately, none of the currently available constructions for B-spline-like bases of Tchebycheffian spline spaces is very well suited for their efficient and robust numerical evaluation and manipulation, due to computational complexity and/or numerical instabilities. This drawback has seriously penalized Tchebycheffian splines, so far, in practical applications despite their great potential, and has confined them mostly to the role of an elegant theoretical extension of the polynomial case.The aim of this paper is to formulate an approach that circumvents the aforementioned complexity in working with Tchebycheffian splines.…”