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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

1993
*
IEEE Computer Graphics and Applications
*

As

doi:10.1109/38.180116
fatcat:eivyti4znbhuhgxhhn5k2zpuxy
*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. ...##
###
On multivariate polynomials in Bernstein–Bézier form and tensor algebra

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). ...

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

1992
*
Mathematics of Computation
*

We present

doi:10.1090/s0025-5718-1992-1134724-1
fatcat:pqlgmcaokjetrh7f3xtlf6pkty
*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). ...##
###
A Blossoming Development of Splines

2006
*
Synthesis Lectures on Computer Graphics and Animation
*

Bézier/B-

doi:10.2200/s00041ed1v01200607cgr001
fatcat:5shteg26izdvdj4ndz2xb6xv2m
*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. ...##
###
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

1993
*
Acta Numerica
*

ACKNOWLEDGEMENT It is

doi:10.1017/s0962492900002348
fatcat:hlx3v6jj7jgyjbypfef5fbaqca
*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. ...##
###
Computation of Polytopic Invariants for Polynomial Dynamical Systems using Linear Programming
[article]

2012
*
arXiv
*
pre-print

An invariant set

arXiv:1012.1256v3
fatcat:6olyzso6tba47djlxcusgeqbpy
*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. ...##
###
On the completeness of hierarchical tensor-product B-splines

2014
*
Journal of Computational and Applied Mathematics
*

We prove that the connected components

doi:10.1016/j.cam.2014.04.001
fatcat:24iic6q6kjekjdqolldtgf6ngy
*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. ...##
###
Developments in bivariate spline interpolation

2000
*
Journal of Computational and Applied Mathematics
*

The aim

doi:10.1016/s0377-0427(00)00346-0
fatcat:42yhs4ss7zapzng54jhwewst6m
*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. ...##
###
Construction of C^2 cubic splines on arbitrary triangulations
[article]

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. ...

##
###
The Bernstein polynomial basis: A centennial retrospective

2012
*
Computer Aided Geometric Design
*

Originally introduced by Sergei Natanovich Bernstein to facilitate

doi:10.1016/j.cagd.2012.03.001
fatcat:eiqucogpb5gh3lh5gnoj7p32cq
*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. ...
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