Multivariate piecewise polynomials

This article was supposed to be on ‘multivariate splines». 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 piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

[1]  W. Dahmen,et al.  Approximation theory VII , 1994 .

[2]  R. DeVore,et al.  Approximation from Shift-Invariant Subspaces of L 2 (ℝ d ) , 1994 .

[3]  Larry L. Schumaker,et al.  The generic dimension of the space of C 1 splines of degree d ≥8 on tetrahedral decompositions , 1993 .

[4]  R. Jia,et al.  Approximation by multiinteger translates of functions having global support , 1993 .

[5]  C. D. Boor,et al.  A sharp upper bound on the approximation order of smooth bivariate PP functions , 1993 .

[6]  C. D. Boor,et al.  Box splines , 1993 .

[7]  Carl de Boor,et al.  Approximation Order without Quasi-Interpolants , 1993 .

[8]  C. D. Boor,et al.  Fourier analysis of the approximation power of principal shift-invariant spaces , 1992 .

[9]  C. Micchelli,et al.  Blossoming begets B -spline bases built better by B -patches , 1992 .

[10]  H. Seidel.  Representing piecewise polynomials as linear combinations of multivariate B-splines , 1992 .

[11]  Larry L. Schumaker,et al.  On dimension and existence of local bases for multivariate spline spaces , 1992 .

[12]  Amos Ron,et al.  The L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a Radial Basis Function , 1992 .

[13]  L. Schumaker,et al.  The L 2 -approximation Orders of Principal Shift-invariant Spaces Generated by a Radial Basis Function X1. Introduction Numerical Methods of Approximation Theory 1 , 1992 .

[14]  Ronald A. DeVore,et al.  The Structure of Nitely Generated Shift-invariant Spaces in L 2 (ir D ) , 1992 .

[15]  R. DeVore,et al.  The Structure of Finitely Generated Shift-Invariant Spaces in , 1992 .

[16]  R. Beatson,et al.  Quasi-interpolation in the Absence of Polynomial Reproduction , 1992 .

[17]  R. Jia.  Approximation by Multivariate Splines: an Application of Boolean Methods , 1992 .

[18]  H. Seidel.  Symmetric recursive algorithms for surfaces: B-patches and the de boor algorithm for polynomials over triangles , 1991 .

[19]  Allan Pinkus,et al.  Progress in Approximation Theory , 1991 .

[20]  Lauren L. Rose,et al.  A dimension series for multivariate splines , 1991, Discret. Comput. Geom..

[21]  A. Ron.  A characterization of the approximation order for multivariate spline spaces , 1991 .

[22]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[23]  C. Micchelli,et al.  Computation of Curves and Surfaces , 1990 .

[24]  C. Chui,et al.  Approximation Theory VI , 1990 .

[25]  A. Ron,et al.  Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems , 1990 .

[26]  A. Le Méhauté,et al.  A Finite Element Approach to Surface Reconstruction , 1990 .

[27]  C. D. Boor,et al.  Quasiinterpolants and Approximation Power of Multivariate Splines , 1990 .

[28]  Lyle Ramshaw,et al.  Blossoms are polar forms , 1989, Comput. Aided Geom. Des..

[29]  L. Billera.  The algebra of continuous piecewise polynomials , 1989 .

[30]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design , 1989 .

[31]  L. Billera,et al.  Gro¨bner basis methods for multivariate splines , 1989 .

[32]  Dieter Lasser,et al.  Grundlagen der geometrischen Datenverarbeitung , 1989 .

[33]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[34]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[35]  Klaus Höllig,et al.  Approximation power of smooth bivariate PP functions , 1988 .

[36]  L. Billera.  Homology of smooth splines: generic triangulations and a conjecture of Strang , 1988 .

[37]  Larry L. Schumaker,et al.  Topics in Multivariate Approximation , 1987 .

[38]  Thomas A. Grandine,et al.  The computational cost of simplex spline functions , 1987 .

[39]  Charles K. Chui,et al.  Cardinal Interpolation by Multivariate Splines , 1987 .

[40]  Charles K. Chui,et al.  On multivariate Vertex splines and Applications , 1987, Topics in Multivariate Approximation.

[41]  Larry L. Schumaker,et al.  A Bibliography of multivariate Approximation , 1987, Topics in Multivariate Approximation.

[42]  Norbert Luscher,et al.  Die Bernstein-Bézier-Technik in der Methode der finiten Elemente , 1987 .

[43]  Gerald Farin,et al.  Geometric modeling : algorithms and new trends , 1987 .

[44]  C. D. Boor,et al.  B-Form Basics. , 1986 .

[45]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[46]  Ronald A. DeVore,et al.  Partitions of Unity and Approximation , 1985 .

[47]  Wolfgang Dahmen,et al.  Some results on box splines , 1984 .

[48]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

[49]  C. Chui,et al.  Approximation Theory IV , 1984 .

[50]  Larry L. Schumaker,et al.  Bounds on the dimension of spaces of multivariate piecewise polynomials , 1984 .

[51]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[52]  Ron Goldman,et al.  Subdivision algorithms for Bézier triangles , 1983 .

[53]  Klaus Höllig,et al.  Approximation order from bivariate ¹-cubics: a counterexample , 1983 .

[54]  Ronald A. DeVore,et al.  Approximation by smooth multivariate splines , 1983 .

[55]  C. Micchelli,et al.  Recent Progress in multivariate splines , 1983 .

[56]  W. Schempp,et al.  Multivariate Approximation Theory IV , 1989 .

[57]  C. Micchelli,et al.  On the Linear Independence of Multivariate B-Splines, I. Triangulations of Simploids , 1982 .

[58]  Klaus Höllig,et al.  Recurrence Relations for Multivariate B-Splines. , 1982 .

[59]  M. Powell,et al.  Approximation theory and methods , 1984 .

[60]  S. L. Lee,et al.  Spline approximation operators of Bernstein-Schoenberg type in one and two variables , 1981 .

[61]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[62]  Wolfgang Dahmen,et al.  Multivariate B-Splines — Recurrence Relations and Linear Combinations of Truncated Powers , 1979 .

[63]  L. Schumaker.  On the Dimension of Spaces Of Piecewise Polynomials in Two Variables , 1979 .

[64]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[65]  Charles A Micchelli,et al.  A Constructive Approach to Kergin Interpolation in R(k). , 1978 .

[66]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[67]  C. Chui,et al.  Approximation Theory II , 1976 .

[68]  C. D. Boor,et al.  Splines as linear combinations of B-splines. A Survey , 1976 .

[69]  Gilbert Strang,et al.  The dimension of piecewise polynomial spaces, and one-sided approximation , 1974 .

[70]  G. Strang.  Piecewise polynomials and the finite element method , 1973 .

[71]  A. Ženíšek.  Polynomial approximation on tetrahedrons in the finite element method , 1973 .

[72]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[73]  A. Ženíšek.  Interpolation polynomials on the triangle , 1970 .

[74]  Plerre Bézier.  Emploi des machines a commande numérique , 1970 .

[75]  I. J. Schoenberg,et al.  Cardinal interpolation and spline functions , 1969 .

[76]  T. J. Rivlin,et al.  Approximation of Functions , 1967 .

[77]  I. J. Schoenberg,et al.  On Pólya frequency functions IV: The fundamental spline functions and their limits , 1966 .

[78]  W. E. Milne.  Numerical Solution of Differential Equations , 1953 .

[79]  I. J. Schoenberg.  Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae , 1946 .

[80]  R. Courant.  Variational methods for the solution of problems of equilibrium and vibrations , 1943 .

[81]  N. E. Nörlund.  Vorlesungen über Differenzenrechnung , 1924 .