ECT-B-splines defined by generalized divided differences

Mühlbach, G.

doi:10.1016/j.cam.2005.03.040

Cited by 20 publications

(22 citation statements)

References 7 publications

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“…If ever the new sequence B * , ∈ Z, is known to satisfy the positivity property (27), then the decomposition relations (28) ensure that so does the initial sequence B , ∈ Z. It is thus sufficient to check that the positivity property (27) is satisfied by the sequence B , ∈ Z, when each knot t k , k ∈ Z, has multiplicity m k = n + 1.…”

Section: K) Is a B-spline-like Basis In The Space Ds(e K)mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 97%

“…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.…”

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confidence: 99%

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

Mazure

2011

Numer. Math.

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International audienceIn the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations

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Section: K) Is a B-spline-like Basis In The Space Ds(e K)mentioning

confidence: 99%

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confidence: 97%

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

Mazure

2011

Numer. Math.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Constructing totally positive piecewise Chebyshevian B-spline bases

Mazure

2018

Journal of Computational and Applied Mathematics

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“…They are studied from a blossom point of view by Mazure [9,10], Mazure and Laurent [11], Mazure and Pottmann [12], Pottmann [18] and more recently by Prautzsch [19], and from a constructive point of view by Mühlbach [15][16][17]. In this paper we consider cardinal ECT-splines generalizing an approach of Barry et al [2], see also [6], discussing cardinal polynomial splines determined by connection matrices.…”

Section: Introductionmentioning

confidence: 97%

Cardinal ECT-splines

Tang

Mühlbach²

2005

Numer Algor

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Cardinal ECT-spline curves are generated from one ECT-system of order n which is shifted by integer translations via one connection matrix. If this matrix is nonsingular, lower triangular and totally positive, there exists an ECT-B-spline function N n 0 (x) having minimal compact support [0, n] whose integer translates span the cardinal ECT-spline space. This B-spline is computed explicitly piece by piece. Involved is the characteristic polynomial of a certain matrix which is the product of a matrix related to the connection matrix and of the generalized Taylor matrix of the basic ECT-system. This approach extends results for polynomial cardinal splines via connection matrices [6] to the more general setting of cardinal ECT-splines. The method is illustrated by two examples based on ECT-systems of rational functions with prescribed poles. Also, a Green's function involved is expressed explicitly as an ECT-B-splines series.

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ECT-B-splines defined by generalized divided differences

Cited by 20 publications

References 7 publications

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

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

Constructing totally positive piecewise Chebyshevian B-spline bases

Cardinal ECT-splines

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