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Page 207 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
A nonexistence case is also obtained. Craciun Iancu (Cluj-Napoca) 93a:41023 93a:41019 41A15 65D07 Lai, Ming Jun (1-UT) A characterization theorem of multivariate splines in blossoming form. Comput.  ...  This principle, called the blossoming principle, is applied to the study of multivariate splines in this paper.  ... 

Page 3939 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
Summary: “In the present paper we obtain a few inclusion the- orems for the convolution of N6érlund methods in the form (N, rn) © (N, pn) *(N, gn) in complete, nontrivially valued, non- Archimedean fields  ...  Powell, A review of algorithms for thin plate spline interpolation in two dimensions (303-322); Paul Sablonniére, B-splines on uniform meshes of the plane (323-338); J.  ... 

Page 4504 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
It is worth mentioning that, in another related work, the author used discrete multivariate spline methods in order to prove a conjecture of R.  ...  He characterizes com- pletely the linear independence property of the shifts of a discrete box spline.  ... 

An introduction to polar forms

H.-P. Seidel
1993 IEEE Computer Graphics and Applications  
As a consequence, we obtain a simple new labeling scheme for B ezier and B-spline curves and surfaces that allows us to label control points in a consistent and meaningful way.  ...  The presentation concludes with a survey of some recent new results that were obtained using polar forms.  ...  Some of its main features are summarized in the following theorem: Theorem 9.1 (Multivariate B-splines) Let F(u) = P I N I ijk (u)c I ijk be a multivariate B-spline surface.  ... 
doi:10.1109/38.180116 fatcat:eivyti4znbhuhgxhhn5k2zpuxy

On multivariate polynomials in Bernstein–Bézier form and tensor algebra

Hendrik Speleers
2011 Journal of Computational and Applied Mathematics  
In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein-Bézier form.  ...  The Bernstein-Bézier representation of polynomials is a very useful tool in computer aided geometric design.  ...  Acknowledgement Hendrik Speleers is a Postdoctoral Fellow of the Research Foundation Flanders (Belgium).  ... 
doi:10.1016/j.cam.2011.04.032 fatcat:qkrqkzqv6jbytbeeut576qpcii

Blossoming begets $B$-spline bases built better by $B$-patches

Wolfgang Dahmen, Charles A. Micchelli, Hans-Peter Seidel
1992 Mathematics of Computation  
We present a general scheme for constructing a collection of multivariate S-splines with k-l continuous derivatives whose linear span contains all polynomials of degree at most k .  ...  The concept of symmetric recursive algorithm leads to new, sdimensional spline spaces.  ...  In this situation, cL(xl, ... , xl) is the polar form or blossom of the polynomial CL(x).  ... 
doi:10.1090/s0025-5718-1992-1134724-1 fatcat:pqlgmcaokjetrh7f3xtlf6pkty

A Blossoming Development of Splines

Stephen Mann
2006 Synthesis Lectures on Computer Graphics and Animation  
Bézier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define the shape of the surface.  ...  Blossoming is used to explore both Bézier and B-spline curves, and in particular to investigate continuity properties, change of basis algorithms, forward differencing, B-spline knot multiplicity, and  ...  We start by defining a few terms used in the blossoming theorem. Definition 2.3.  ... 
doi:10.2200/s00041ed1v01200607cgr001 fatcat:5shteg26izdvdj4ndz2xb6xv2m

Page 8440 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
et ses conséquences [A characterization of measured geodesic pleating laminations of hyperbolic manifolds, and its consequences] (103 115); Constantin Vernicos, Formes harmoniques de longueur con- stante  ...  Leviatan, On measuring the efficiency of kernel operators in L,(IR“) (53-65); K. Jetter and G. Zimmermann [Georg Zimmermann], Polynomial reproduction in subdivision (67-86); A. Kouibia and M.  ... 

Page 2197 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
Vinacua, Modeling closed surfaces: a comparison of ex- isting methods (29-42); Guido Brunnett, A new characterization of plane elastica (43-56); Paul de Faget de Casteljau, POLynomi- als, POLar forms,  ...  Seidel, Representing piecewise polynomials as linear combinations of multivariate B- splines (559-566); Gadiel Seroussi and Brian A.  ... 

Multivariate piecewise polynomials

C. de Boor
1993 Acta Numerica  
ACKNOWLEDGEMENT It is a pleasure to thank Martin Buhmann for his many constructive comments on a draft of this paper.  ...  An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a`multivariate spline', resulted in the answer that a multivariate spline is a possibly smooth  ...  In contrast, the simplex splines were expected to be the multivariate equivalent of the general univariate B-spline, of use in the understanding and handling of arbitrary multivariate spline spaces.  ... 
doi:10.1017/s0962492900002348 fatcat:hlx3v6jj7jgyjbypfef5fbaqca

Computation of Polytopic Invariants for Polynomial Dynamical Systems using Linear Programming [article]

Mohamed Amin Ben Sassi, Antoine Girard
2012 arXiv   pre-print
An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given instant, it will remain in the set forever in the future.  ...  Polytopic invariants for polynomial systems can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes.  ...  Multi-affine functions form a particular class of multivariate polynomials.  ... 
arXiv:1012.1256v3 fatcat:6olyzso6tba47djlxcusgeqbpy

On the completeness of hierarchical tensor-product B-splines

Dominik Mokriš, Bert Jüttler, Carlotta Giannelli
2014 Journal of Computational and Applied Mathematics  
We prove that the connected components of the associated set of tensor-product B-splines, whose support intersects the multi-cell domain, form a basis of this spline space.  ...  Given a grid in R d , consisting of d bi-infinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensor-product spline  ...  The dimension of the spline space is then described in Theorem A.8. Lemma A.6.  ... 
doi:10.1016/j.cam.2014.04.001 fatcat:24iic6q6kjekjdqolldtgf6ngy

Developments in bivariate spline interpolation

G. Nürnberger, F. Zeilfelder
2000 Journal of Computational and Applied Mathematics  
The aim of this survey is to describe developments in the ÿeld of interpolation by bivariate splines.  ...  We summarize results on the dimension and the approximation order of bivariate spline spaces, and describe interpolation methods for these spaces. Moreover, numerical examples are given.  ...  We also note that many papers on so-called multivariate simplex splines and multivariate box splines exist in the literature.  ... 
doi:10.1016/s0377-0427(00)00346-0 fatcat:42yhs4ss7zapzng54jhwewst6m

Construction of C^2 cubic splines on arbitrary triangulations [article]

Tom Lyche, Carla Manni, Hendrik Speleers
2021 arXiv   pre-print
In this paper, we address the problem of constructing C^2 cubic spline functions on a given arbitrary triangulation 𝒯. To this end, we endow every triangle of 𝒯 with a Wang-Shi macro-structure.  ...  The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for C^2 joins  ...  From the proof of Theorem 3 it follows that we can formulate a Hermite interpolation problem to characterize any spline in S 2 3 (∆ WS 3 ). Corollary 4.  ... 
arXiv:2110.07907v1 fatcat:i5qlrbl6czg5fgt3vx36rppusq

The Bernstein polynomial basis: A centennial retrospective

Rida T. Farouki
2012 Computer Aided Geometric Design  
Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leisurely convergence rate of Bernstein polynomial approximations to  ...  With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating  ...  Acknowledgements The idea for this survey originated among discussions between Ron Goldman, Hartmut Prautzsch, and others at a meeting in Schloss Dagstuhl in May 2011.  ... 
doi:10.1016/j.cagd.2012.03.001 fatcat:eiqucogpb5gh3lh5gnoj7p32cq
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