Topics in Multivariate Approximation Theory.

Abstract : Lectures delivered at the S.R.C. Numerical Analysis Summer School and Workshop at the University of Lancaster, England, July 19 - August 20, 1981. Topics include tensor products, multivariate polynomial interpolation, esp. Kergin Interpolation, and the recent developments of multivariate B-splines. (Author)

[1]  J. Rice.  The approximation of functions , 1964 .

[2]  P. G. Ciarlet,et al.  General lagrange and hermite interpolation in Rn with applications to finite element methods , 1972 .

[3]  K. Chung,et al.  On Lattices Admitting Unique Lagrange Interpolations , 1977 .

[4]  J. H. Bramble,et al.  Bounds for a class of linear functionals with applications to Hermite interpolation , 1971 .

[5]  Wolfgang Dahmen,et al.  Polynomials as linear combinations of multivariateB-splines , 1979 .

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

[7]  F. di Gugliemlo.  Construction d’approximations des espaces de sobolev sur des reseaux en simplexes , 1969 .

[8]  G. Strang,et al.  Fourier Analysis of the Finite Element Method in Ritz-Galerkin Theory , 1969 .

[9]  J. Bramble,et al.  Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation , 1970 .

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

[11]  E. Cheney,et al.  On the existence and characterization of minimal projections. , 1972 .

[12]  Klaus Höllig,et al.  B-splines from parallelepipeds , 1982 .

[13]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

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

[15]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[16]  Klaus Höllig.  A remark on multivariate B-splines☆ , 1981 .

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

[18]  Ren-Hong Wang,et al.  Multivariate B-splines on triangulated rectangles☆ , 1983 .

[19]  Robert E. Barnhill,et al.  Representation and Approximation of Surfaces , 1977 .

[20]  M. Gasca,et al.  On Lagrange and Hermite interpolation in Rk , 1982 .

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

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

[23]  Paul Kergin,et al.  A natural interpolation of Ck functions , 1980 .

[24]  Wolfgang Dahmen,et al.  Multidimensional Spline Approximation , 1980 .

[25]  P. G. Ciarlet,et al.  Interpolation theory over curved elements, with applications to finite element methods , 1972 .

[26]  Hakop Hakopian,et al.  Multivariate Spline Functions, B-Spline Basis and Polynomial Interpolations , 1982 .

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

[28]  Carl de Boor,et al.  Efficient Computer Manipulation of Tensor Products , 1979, TOMS.

[29]  Ren-Hong Wang,et al.  Multivariate spline spaces , 1983 .

[30]  Charles A. Micchelli,et al.  A formula for Kergin interpolation in Rk , 1980 .

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

[32]  John R. Rice,et al.  An adaptive algorithm for multivariate approximation giving optimal convergence rates , 1979 .

[33]  G. Lorentz.  Approximation of Functions , 1966 .

[34]  Wolfgang Dahmen,et al.  On limits of multivariateB-splines , 1981 .

[35]  E. Cheney.  Introduction to approximation theory , 1966 .

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

[37]  Jean Duchon,et al.  Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces , 1976 .

[38]  C. Micchelli.  On a numerically efficient method for computing multivariate B-splines , 1979 .

[39]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[40]  A. Kolmogoroff,et al.  Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse , 1936 .

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

[42]  L. Schumaker.  Fitting surfaces to scattered data , 1976 .

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

[44]  N. I. Achieser.  Vorlesungen über Approximationstheorie , 1967 .

[45]  J. Duchon.  Spline minimizing rotation-invariant seminorms in Sobolev spaces , 1977 .

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

[47]  Wolfgang Dahmen,et al.  Approximation by linear combinations of multivariate B-splines , 1981 .

[48]  Wolfgang Dahmen.  Adaptive approximation by multivariate smooth splines , 1982 .

[49]  Wolfgang Dahmen,et al.  On Multivariate B-Splines , 1980 .

[50]  J. Meinguet.  Multivariate interpolation at arbitrary points made simple , 1979 .

[51]  J. I. Maeztu.  Divided Differences Associated with Reversible Systems in $\mathbb{R}^2 $ , 1982 .

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