Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-12T22:47:57.152Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate piecewise polynomials

Published online by Cambridge University Press:  07 November 2008

C. de Boor
C. de Boor
Affiliation:
Center for Mathematical SciencesUniversity of Wisconsin-MadisonMadison WI 53705USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Acta Numerica , Volume 2 , January 1993 , pp. 65 - 109
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alfeld, P., Schumaker, L.L. and Sirvent, Maritza (1992), ‘On dimension and existence of local bases for multivariate spline spaces’, J. Approx. Theory 70, 243–264.CrossRefGoogle Scholar
Alfeld, P., Whiteley, W. and Schumaker, L.L. (199x), ‘The generic dimension of the space of C1 splines of degree d ≥ 8 on tetrahedral decompositions’, SIAM J. Numer. Anal. xx, xxx–xxx.Google Scholar
Beatson, R. K. and Light, W. A. (1992), ‘Quasi-interpolation in the absence of polynomial reproduction’, Univ. of Leicester Mathematics Techn. Report 1992/15, May.CrossRefGoogle Scholar
Bézier, P.E. (1970), Emploi des Machines à Commande Numérique, Masson et Cie (Paris).Google Scholar
Beziér, P.E. (1977), ‘Essai de définition numérique des courbes et des surfaces expérimentale’, dissertation, Univ. Paris.Google Scholar
Billera, L.J. (1988), ‘Homology of smooth splines: generic triangulations and a conjecture of Strang’, Trans. Amer. Math. Soc. 310, 325–340.CrossRefGoogle Scholar
Billera, L.J. (1989), ‘The algebra of continuous piecewise polynomials’, Advances of Math. 76(2), 170–183.CrossRefGoogle Scholar
Billera, L.J. and Haas, R. (1987), ‘Homology of divergence-free splines’, Cornell.Google Scholar
Billera, L.J. and Rose, L.L. (1989), ‘Gröbner basis methods for multivariate splines’, in Mathematical Methods in Computer Aided Geometric Design (Lyche, T. and Schumaker, L., eds), Academic Press (New York), 93–104.CrossRefGoogle Scholar
Billera, Louis J. and Rose, Lauren L. (1991), ‘A dimension series for multivariate splines’, Discrete & Computational Geometry 6, 107–128.CrossRefGoogle Scholar
Birkhoff, G. and de Boor, C.R. (1965), ‘Piecewise polynomial interpolation and approximation’, in Approximation of Functions (Garabedian, H.L., ed.), Elsevier (Amsterdam), 164–190.Google Scholar
Boehm, W., Farin, G. and Kahmann, J. (1984), ‘A survey of curve and surface methods in CAGD’, Comput. Aided Geom. Design 1, 1–60.CrossRefGoogle Scholar
de Boor, C. (1976), ‘Splines as linear combinations of B–splines, a survey’, in Approximation Theory II (Lorentz, G. G., Chui, C. K. and Schumaker, L. L., eds), Academic Press (New York), 1–47.Google Scholar
de Boor, C. (1978), A Practical Guide to Splines, Springer Verlag (New York).CrossRefGoogle Scholar
de Boor, C. (1987), ‘B–form basics’, in Geometric Modeling: Algorithms and New Trends (Farin, G. E., ed.), SIAM Publications (Philadelphia), 131–148.Google Scholar
de Boor, C. (1990), ‘Quasiinterpolants and approximation power of multivariate splines’, in Computation of Curves and Surfaces (Dahmen, W., Gasca, M. and Micchelli, C., eds), Kluwer, 313–345.CrossRefGoogle Scholar
de Boor, C. (1992), ‘Approximation order without quasi-interpolants’, in Approximation Theory VII (Cheney, E. W., Chui, C. and Schumaker, L., eds), Academic Press (New York), xxx–xxx.Google Scholar
de Boor, C. and DeVore, R. (1983), ‘Approximation by smooth multivariate splines’, Trans. Amer. Math. Soc. 276, 775–788.CrossRefGoogle Scholar
de Boor, C. and DeVore, R. (1985), ‘Partitions of unity and approximation’, Proc. Amer. Math. Soc. 93, 705–709.CrossRefGoogle Scholar
de Boor, C. and Höllig, K. (1982), ‘Recurrence relations for multivariate B-splines’, Proc. Amer. Math. Soc. 85, 397–400.CrossRefGoogle Scholar
de Boor, C. and Höllig, K. (1983), ‘Approximation order from bivariate C1-cubics: a counterexample’, Proc. Amer. Math. Soc. 87, 649–655.CrossRefGoogle Scholar
de Boor, C. and Höllig, K. (1988), ‘Approximation power of smooth bivariate pp functions’, Math. Z. 197, 343–363.CrossRefGoogle Scholar
de Boor, C. and Jia, Rong-Qing (199x), ‘A sharp upper bound on the approximation order of smooth bivariate pp functions’, J. Approx. Theory XX, xxx–xxx.Google Scholar
de Boor, C. and Ron, A. (1992), ‘Fourier analysis of the approximation power of principal shift-invariant spaces’, Constr. Approx. 8, 427–462.CrossRefGoogle Scholar
de Boor, C., DeVore, R. and Ron, A. (1991), ‘Approximation from shift-invariant subspaces of L2(ℝd)’, CMS-TSR 92–2, University of Wisconsin (Madison).CrossRefGoogle Scholar
de Boor, C., DeVore, R. and Ron, A. (1992a), ‘The structure of finitely generated shift-invariant spaces in L2(ℝd)’, CMS-TSR 92–8, University of Wisconsin (Madison).CrossRefGoogle Scholar
de Boor, C., Höllig, K. and Riemenschneider, S.D. (1992), Box Splines, Springer-Verlag (Berlin).Google Scholar
de Casteljau, P. (1963), ‘Courbes et Surfaces à Pôles’, André Citroën Automobile SA (Paris).Google Scholar
de Casteljau, P. (1985), Formes à Pôles, Hermes (Paris).Google Scholar
Chui, C.K. (1988), Multivariate Splines, CBMS-NSF Reg. Conf. Series in Appl. Math., Vol. 54, SIAM (Philadelphia).CrossRefGoogle Scholar
Chui, C.K. and Lai, M.J. (1987), ‘On multivariate vertex splines and applications’, in Topics in Multivariate Approximation (Chui, C. K., Schumaker, L. L. and Utreras, F., eds), Academic Press (New York), 19–36.CrossRefGoogle Scholar
Chui, C.K., Jetter, K. and Ward, J.D. (1987), ‘Cardinal interpolation by multivariate splines’, Math. Comput. 48, 711–724.CrossRefGoogle Scholar
Courant, R. (1943), ‘Variational methods for the solution of problems in equilibrium and vibrations’, Bull. Amer. Math. Soc. 49, 1–23.CrossRefGoogle Scholar
Curry, H.B. and Schoenberg, I.J. (1966), ‘On the Pólya frequency functions IV: the fundamental spline functions and their limits’, J. Analyse Math. 17, 71–107.CrossRefGoogle Scholar
Dahmen, W. (1979), ‘Multivariate B-splines—Recurrence relations and linear combinations of truncated powers’, in Multivariate Approximation Theory (Schempp, W. and Zeller, K., eds), Birkhäuser (Basel), 64–82.CrossRefGoogle Scholar
Dahmen, W. and Micchelli, C.A. (1982), ‘On the linear independence of multivariate B-splines I. Triangulations of simploids’, SIAM J. Numer. Anal. 19, 992–1012.CrossRefGoogle Scholar
Dahmen, W. and Micchelli, C.A. (1983), ‘Recent progress in multivariate splines’, in Approximation Theory IV (Chui, C., Schumaker, L. and Ward, J., eds), Academic Press (New York), 27–121.Google Scholar
Dahmen, W. and Micchelli, C.A. (1984), ‘Some results on box splines’, Bull. Amer. Math. Soc. 11, 147–150.CrossRefGoogle Scholar
Dahmen, W., Micchelli, C.A. and Seidel, H.-P. (1992), ‘Blossoming begets B-spline bases built better by B-patches’, Math. Comput. 59, 97–115.Google Scholar
Dyn, N. and Ron, A. (1990), ‘Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems’, Trans. Amer. Math. Soc. 319, 381–404.CrossRefGoogle Scholar
Farin, G. (1977), ‘Konstruktion und Eigenschaften von Bézier-Kurven und Bézier Flächen’, Diplom-Arbeit, Univ. Braunschweig.Google Scholar
Farin, G. (1986), ‘Triangular Bernstein-Bézier patches’, Comput. Aided Geom. Design 2, 83–127.CrossRefGoogle Scholar
Farin, G. (1988), Curves and Surfaces for Computer Aided Geometric Design, Academic Press (New York).Google Scholar
Faux, Ivor D. and Pratt, Michael J. (1979), Computational Geometry for Design and Manufacture, Ellis Horwood (Chichester).Google Scholar
Franke, R. and Schumaker, L.L. (1987), ‘A bibliography of multivariate approximation’, in Topics in Multivariate Approximation (Chui, C. K., Schumaker, L. L. and Utreras, F., eds), Academic Press (New York), 275–335.CrossRefGoogle Scholar
Goldman, R.N. (1983), ‘Subdivision algorithms for Bézier triangles’, Computer-Aided Design 15, 159–166.CrossRefGoogle Scholar
Goodman, T.N. and Lee, S.L. (1981), ‘Spline approximation operators of Bernstein–Schoenberg type in one and two variables’, J. Approx. Theory 33, 248–263.CrossRefGoogle Scholar
Grandine, T.A. (1986), ‘The computational cost of simplex spline functions’, MRC Rpt. 2926, Univ. Wis. (Madison WI).Google Scholar
Höllig, K. (1982), ‘Multivariate splines’, SIAM J. Numer. Anal. 19, 1013–1031.CrossRefGoogle Scholar
Höllig, K. (1986), ‘Multivariate splines’, in Approximation Theory, Proc. Symp. Appl. Math. 36 (de Boor, C., ed.), Amer. Math. Soc. (Providence, RI), 103–127.CrossRefGoogle Scholar
Hoschek, J. and Lasser, D. (1989), Grundlagen der geometrischen Datenverarbeitung, B.G. Teubner (Stuttgart).CrossRefGoogle Scholar
Jia, Rong-Qing (1989), ‘Approximation order of translation invariant subspaces of functions’, in Approximation Theory VI (Chui, C., Schumaker, L. and Ward, J., eds), Academic Press (New York), 349–352.Google Scholar
Jia, Rong-Qing (1992), ‘Approximation by multivariate splines: an application of Boolean methods’, ms.CrossRefGoogle Scholar
Jia, Rong-Qing and Lei, Junjiang (1991), ‘Approximation by multiinteger translates of functions having global support’, J. Approx. Theory X, xxx–xxx.Google Scholar
Méhauté, A. Le (1990), ‘A finite element approach to surface reconstruction’, in Computation of Curves and Surfaces (Dahmen, W., Gasca, M. and Micchelli, C., eds), Kluwer, 237–274.CrossRefGoogle Scholar
Lorentz, G.G. (1953), Bernstein Polynomials, Toronto Press (Toronto).Google Scholar
Luscher, N. (1987), ‘Die Bernstein-Bézier Technik in der Methode der finiten Elemente’, dissertation, Univ. Braunschweig.Google Scholar
Micchelli, C.A. (1980), ‘A constructive approach to Kergin interpolation in ℝk: multivariate B-splines and Lagrange interpolation’, Rocky Mountain J. Math. 10, 485–97.CrossRefGoogle Scholar
Morgan, J. and Scott, R. (1990), ‘The dimension of the space of C1 piecewise polynomials’, Research Report UH/MD-78, University of Houston.Google Scholar
Nörlund, N.E. (1924), Vorlesungen über Differenzenrechnung, Grundlehren Vol. XIII, Springer (Berlin).CrossRefGoogle Scholar
Powell, M.J.D. (1981), Approximation Theory and Methods, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Ramshaw, L. (1987), ‘Blossoming: a connect-the-dots approach to splines’, Techn. Rep., Digital Systems Research Center (Palo Alto).Google Scholar
Ramshaw, L. (1989), ‘Blossoms are polar forms’, Comput. Aided Geom. Design 6, 23–32.CrossRefGoogle Scholar
Ron, A. (1991), ‘A characterization of the approximation order of multivariate spline spaces’, Studia Math. 98, 73–90.CrossRefGoogle Scholar
Ron, A. (1992), ‘The L2-approximation orders of principal shift-invariant spaces generated by a radial basis function’, in Numerical Methods in Approximation Theory, ISNM 105 (Braess, D., Schumaker, L.L., eds), Birkhäuser (Basel), 245–268.CrossRefGoogle Scholar
Schoenberg, I. J. (1946), ‘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 formulas’, Quart. Appl. Math. 4, 45–99.CrossRefGoogle Scholar
Schoenberg, I.J. (1965), ‘Letter to Philip J. Davis’, dated May 31.Google Scholar
Schoenberg, I.J. (1969), ‘Cardinal interpolation and spline functions’, J. Approx. Theory 2, 167–206.CrossRefGoogle Scholar
Schumaker, L.L. (1979), ‘On the dimension of spaces of piecewise polynomials in two variables’, in Multivariate Approximation Theory (Schempp, W. and Zeller, K., eds), Birkhäuser (Basel), 396–412.CrossRefGoogle Scholar
Schumaker, L.L. (1981), Spline Functions: Basic Theory, Wiley (New York).Google Scholar
Schumaker, L.L. (1984), ‘Bounds on the dimension of spaces of multivariate piecewise polynomials’, Rocky Mountain J. Math. 14, 251–264.CrossRefGoogle Scholar
Schumaker, L.L. (1988), ‘Constructive aspects of bivariate piecewise polynomials’, in The Mathematics of Finite Elements an Applications VI (Whiteman, J.R., ed.), Academic Press (London), 513–520.Google Scholar
Schumaker, L.L. (1991), ‘Recent progress on multivariate splines’, in Mathematics of Finite Elements and Applications VII (Whiteman, J., ed.), Academic Press (London), 535–562.Google Scholar
Seidel, H.-P. (1991), ‘Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles’, Constr. Approx. 7, 257–279.CrossRefGoogle Scholar
Seidel, H.-P. (1991), ‘Representing piecewise polynomials as linear combinations of multivariate B-splines’, ms.CrossRefGoogle Scholar
Stein, E. and Weiss, G. (1971), Introduction to Fourier analysis on Euclidean spaces, Princeton University Press (Princeton).Google Scholar
Strang, G. (1973), ‘Piecewise polynomials and the finite element method’, Bull. Amer. Math. Soc. 79, 1128–1137.CrossRefGoogle Scholar
Strang, G. (1974), ‘The dimension of piecewise polynomials, and one-sided approximation’, in Numerical Solution of Differential Equations (Watson, G. A., ed.), Springer (Berlin), 144–152.CrossRefGoogle Scholar
Strang, G. and Fix, G. (1973), ‘A Fourier analysis of the finite element variational method’, in Constructive Aspects of Functional Analysis (Geymonat, G., ed.), C.I.M.E. II Ciclo 1971, 793–840.Google Scholar
Whiteley, W. (1991), ‘A matrix for splines’, in Progress in Approximation Theory (Nevai, P. and Pinkus, A., eds.), Academic Press (Boston), 821–828.Google Scholar
Ženišek, A. (1970), ‘Interpolation polynomials on the triangle’, Numer. Math. 15, 283–296.CrossRefGoogle Scholar
Ženišek, A. (1973), ‘Polynomial approximation on tetrahedrons in the finite element method’, J. Approx. Theory 7, 334–351.CrossRefGoogle Scholar
Ženišek, A. (1974), ‘A general theorem on triangular Cm elements’, Rev. Française Automat. Informat. Rech. Opér., Anal. Numer. 22, 119–127.Google Scholar