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Topics in multivariate approximation theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 965)

Keywords

  • Tensor Product
  • Polynomial Interpolation
  • Normed Linear Space
  • Linear Projector
  • Hermite Interpolation

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References

  • N. I. Akhiezer [1967], Vorlesungen über Approximationstheorie, 2., verbesserte Auflage, Akademie-Verlag, Berlin; appeared also as Theory of Approximation, F. Ungar Publ., New York, 1956.

    MATH  Google Scholar 

  • R. E. Barnhill [1977], Representation and approximation of surfaces, in Mathematical Software III, J. Rice ed., Academic Press, New York, 69–120.

    CrossRef  Google Scholar 

  • C. de Boor [1976], Splines as linear combinations of B-splines, in Approximation Theory II, G.G. Lorentz, C.K. Chui & L.L. Schumaker eds., Academic Press, 1–47.

    Google Scholar 

  • C. de Boor [1979], Efficient computer manipulation of tensor products, ACM Trans.Math.Software 5, 173–182.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. de Boor & R. DeVore [1981], Approximation by smooth multivariate splines, Math.Research Center TSR #2319. Trans.Amer.Math.Soc., to appear.

    Google Scholar 

  • C. de Boor & K. Höllig [1981], Recurrence relations for multivariate B-splines, Math.Research Center TSR #2215. Proc.Amer.Math.Soc., to appear.

    Google Scholar 

  • C. de Boor & K. Höllig [1982], B-splines from parallelepipeds, Math.Research Center TSR #2320.

    Google Scholar 

  • C. de Boor & J. R. Rice [1979], An adaptive algorithm for multivariate approximation giving optimal convergence rates, J.Approx. Theory 25, 337–359.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • J. H. Bramble & S. R. Hilbert [1970], Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J.Numer.Anal. 7, 112–124.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • J. H. Bramble & S. R. Hilbert [1971], Bounds for a class of linear functionals with applications to Hermite interpolation, Numer.Math. 16, 362–369.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • E. W. Cheney [1966], Introduction to Approximation Theory, McGraw-Hill, New York.

    MATH  Google Scholar 

  • C. K. Chui & R.-H. Wang [1981]1, Multivariate B-splines on triangulated rectangles, CAT # 6, Center for Approximation Theory, Texas A&M University, College Station, TX.

    Google Scholar 

  • C. K. Chui & R.-H. Wang [1981]2, On a bivariate B-spline basis, CAT # 7.

    Google Scholar 

  • C. K. Chui & R.-H. Wang [1981]3, Multivariate spline spaces, CAT #9.

    Google Scholar 

  • K. C. Chung & T. H. Yao [1977], On lattices admitting unique Lagrange interpolations, SIAM J.Numer.Anal. 14, 735–741.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • P. G. Ciarlet & R. A. Raviart [1972]1, General Lagrange and Hermite interpolation in Rk with applications to finite element methods, Arch.Rat.Mech.Anal. 46, 177–199.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • P. G. Ciarlet & R. A. Raviart [1972]2, Interpolation theory over curved elements, with applications to finite element methods, Computer Methods in Appl.Mech.Eng. 1, 217–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • R. Courant [1943], Variational methods for the solution of problems of equilibrium and vibrations, Bull.Amer.Math.Soc. 49, 1–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • H. B. Curry & I. J. Schoenberg [1966], Pólya frequency functions IV. The fundamental spline functions and their limits, J.d'Anal.Math. 17, 71–107.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. Dahmen [1979]1, Multivariate B-splines — recurrence relations and linear combinations of truncated powers, in Multivariate Approximation Theory, W. Schempp & K. Zeller eds., Birkhäuser, Basel, 64–82.

    CrossRef  Google Scholar 

  • W. Dahmen [1979]2, Polynomials as linear combinations of multivariate B-splines, Math.Z. 169, 93–98.

    CrossRef  MathSciNet  Google Scholar 

  • W. Dahmen [1980]1, On multivariate B-splines, SIAM J.Numer.Anal. 17, 179–191.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. Dahmen [1980]2, Approximation by linear combinations of multivariate B-splines, J.Approx.Theory, to appear.

    Google Scholar 

  • W. Dahmen [1982], Adaptive approximation by multivariate smooth splines, J.Approx.Theory, to appear.

    Google Scholar 

  • W. Dahmen, R. DeVore & K. Scherer [1980], Multi-dimensional spline approximation, SIAM J.Numer.Anal. 17, 380–402.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. Dahmen & C. A. Micchelli [1980], On limits of multivariate B-splines, Math.Research Center TSR # 2114. J.d'Anal.Math. 39 (1981), 256–278.

    CrossRef  MathSciNet  Google Scholar 

  • W. Dahmen & C. A. Micchelli [1982], On the linear independence of multivariate B-splines.I, Triangulation of simploids, SIAM J.Numer.Anal. xx, xxx–xxx.

    MathSciNet  Google Scholar 

  • P. Davis [1963], Interpolation and Approximation, Blaisdell, Waltham MA. Now available from Dover.

    MATH  Google Scholar 

  • J. Duchon [1976], Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, R.A.I.R.O. Analyse numerique 10, 5–12.

    MathSciNet  Google Scholar 

  • J. Duchon [1977], Splines minimizing rotation-invariant seminorms in Sobolev spaces, in Constructive Theory of Functions in Several Variables, Oberwolfach 1976, W. Schempp & K. Zeller eds., Springer-Verlag, Heidelberg, 85–100.

    CrossRef  Google Scholar 

  • G. Fix & G. Strang [1969], Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies in Appl.Math. 48, 265–273.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • R. Franke [1982], Testing methods for scattered data interpolation and some results, Math.Comp. 38, 181–200.

    MathSciNet  MATH  Google Scholar 

  • M. Gasca & J. I. Maeztu [1980], On Lagrange and Hermite interpolation in Rk, Numer.Math., to appear.

    Google Scholar 

  • F. di Guglielmo [1969], Construction d'approximations des espaces de Sobolev sur des reseaux en simplexes, Calcolo 6, 279–331.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Golomb & H. F. Weinberger [1959], Optimal approximation and error bounds, in On Numerical Approximation, R. Langer ed., U. of Wisconsin Press, 117–190.

    Google Scholar 

  • T. N. T. Goodman & S. L. Lee [1981], Spline approximation operators of Bernstein-Schoenberg type in one and two variables, J.Approx.Theory 33, 248–263.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. J. Gordon [1969]1, Spline-blended surface interpolation through curve networks, J.Math.Mech. 18, 931–952.

    MathSciNet  MATH  Google Scholar 

  • W. J. Gordon [1969]2, Distributive lattices and approximation of multivariate functions, in Approximations with special emphasis on spline functions, I. J. Schoenberg ed., Academic Press, New York, 223–277.

    Google Scholar 

  • H. Hakopian [1980], On multivariate B-splines, SIAM J.Numer.Anal. xx, xxx–xxx.

    MathSciNet  Google Scholar 

  • H. Hakopian [1981]1, Les differences divisees de plusieurs variables et les interpolations multidimensionelles de types lagrangien et hermitien, C.R. Acad.Sc.Paris 292, 453–456.

    MathSciNet  MATH  Google Scholar 

  • H. Hakopian [1981]2, Multivariate spline functions, B-spline basis and polynomial interpolations, ms.

    Google Scholar 

  • K. Höllig [1981], A remark on multivariate B-splines, J.Approx.Theory 33, 119–125.

    CrossRef  MathSciNet  Google Scholar 

  • K. Höllig [1982], Multivariate splines, SIAM J.Numer.Anal. xx, xxx–xxx.

    MathSciNet  Google Scholar 

  • P. Kergin [1978], Interpolation of Ck functions, Ph.D. Thesis, University of Toronto, Canada.

    Google Scholar 

  • P. Kergin [1980], A natural interpolation of Ck functions, J.Approx.Theory 29, 278–293.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • A. N. Kolmogorov [1936], Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann.Math. 37, 107–111.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • G. G. Lorentz [1966], Approximation of Functions, Holt, Rinehart & Winston, New York.

    MATH  Google Scholar 

  • J. I. Maeztu [1982], Divided differences associated with reversible systems in R2, SIAM J.Numer.Anal., to appear.

    Google Scholar 

  • J. Meinguet [1979], Multivariate interpolation at arbitrary points made simple, J.Appl.Math.Phys. (ZAMP) 30, 292–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. A. Micchelli [1980], A constructive approach to Kergin interpolation in Rk: multivariate B-splines and Lagrange interpolation, Rocky Mountains J.Math. 10, 485–497.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. A. Micchelli [1979], On a numerically efficient method for computing multivariate B-splines, in Multivariate Approximation Theory, W. Schempp & K. Zeller eds., Birkhäuser, Basel, 211–248.

    CrossRef  Google Scholar 

  • C. A. Micchelli & P. Milman [1980], A formula for Kergin interpolation in Rk, J.Approx.Theory 29, 294–296.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • J. Morgan & R. Scott [1975], The dimension of the space of C1 piecewise polynomials, ms.

    Google Scholar 

  • P. D. Morris & E. W. Cheney [1974], On the existence and characterization of minimal projectors, J.reine angew.Math. 270, 61–76.

    MathSciNet  MATH  Google Scholar 

  • N. E. Nörlund [1924], Vorlesungen über Differenzenrechnung, Springer Grundlehren Bd. 13, Berlin.

    Google Scholar 

  • M. J. D. Powell [1981], Approximation theory and methods, Cambridge University Press.

    Google Scholar 

  • J. R. Rice [1964, 1969], The Approximation of Functions. Vols. I., II., Addison-Wesley, Reading MA.

    Google Scholar 

  • T. Rivlin [1969], In Introduction to the Approximation of Functions, Blaisdell, Waltham MA.

    MATH  Google Scholar 

  • I. J. Schoenberg [1965], letter to Philip J. Davis dated May 31, 1965.

    Google Scholar 

  • I. J. Schoenberg [1973], Cardinal Spline Interpolation, SIAM, Philadelphia PA.

    CrossRef  MATH  Google Scholar 

  • A. Schönhage [1971], Approximationstheorie, de Gruyter, Berlin.

    CrossRef  MATH  Google Scholar 

  • L. L. Schumaker [1976], Fitting surfaces to scattered data, in Approximation Theory II, G.G. Lorentz, C. K. Chui & L. L. Schumaker eds., Academic Press, New York, 203–268.

    Google Scholar 

  • L. L. Schumaker [1979], On the dimension of spaces of piecewise polynomials in two variables, in Multivariate Approximation Theory, W. Schempp & K. Zeller eds., Birkhäuser, Basel, 396–412.

    CrossRef  Google Scholar 

  • G. Strang [1974], The dimension of piecewise polynomials, and one-sided approximation, Springer Verlag Lecture Notes 363, 144–152.

    MathSciNet  MATH  Google Scholar 

  • G. Strang & G. J. Fix [1973], An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs.

    MATH  Google Scholar 

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de Boor, C. (1982). Topics in multivariate approximation theory. In: Turner, P.R. (eds) Topics in Numerical Analysis. Lecture Notes in Mathematics, vol 965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0063200

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  • DOI: https://doi.org/10.1007/BFb0063200

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