A variational approach to splines

Abstract This is an expository paper in which we present an introduction to a variational approach to spline interpolation. We present a sequence of theorems which starts with Holladay's classical result concerning natural cubic splines and culminates in some general abstract results.

[1]  C. Hermite,et al.  Sur la formule d'interpolation de Lagrange. (Extrait d'une lettre de M. Ch. Hermite à M. Borchardt). , 1877 .

[2]  M. Hermite,et al.  Sur la formule d'interpolation de Lagrange , 1878 .

[3]  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 .

[4]  John C. Holladay,et al.  A smoothest curve approximation , 1957 .

[5]  Michael Golomb,et al.  OPTIMAL APPROXIMATIONS AND ERROR BOUNDS , 1958 .

[6]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[7]  Rudolph E. Langer,et al.  On Numerical Approximation , 1959 .

[8]  J L Walsh,et al.  FUNDAMENTAL PROPERTIES OF GENERALIZED SPLINES. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[9]  T. Greville.  INTERPOLATION BY GENERALIZED SPLINE FUNCTIONS , 1964 .

[10]  C. D. Boor,et al.  On splines and their minimum properties , 1966 .

[11]  Marc Atteia,et al.  Etude de certains noyaux et théorie des fonctions "spline" en analyse numérique , 1966 .

[12]  M. Golomb.  SPLINES, N-WIDTHS AND OPTIMAL APPROXIMATIONS. , 1967 .

[13]  R. Varga,et al.  L-Splines , 1967 .

[14]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[15]  I. J. Schoenberg.  On the Ahlberg-Nilson extension of spline interpolation: The g-splines and their optimal properties☆ , 1968 .

[16]  P. Laurent,et al.  A general method for the construction of interpolating or smoothing spline-functions , 1968 .

[17]  L. Schumaker,et al.  On Lg-splines☆ , 1969 .

[18]  T. N. E. Greville,et al.  Theory and applications of spline functions , 1969 .

[19]  J. L. Lagrange,et al.  Oeuvres de Lagrange , 1970 .

[20]  A theory of generalized splines with applications to nonlinear boundary value problems , 1970 .

[21]  R. N. Desmarais,et al.  Interpolation using surface splines. , 1972 .

[22]  Richard B. Holmes,et al.  R-splines in Banach spaces: I. Interpolation of linear manifolds☆ , 1972 .

[23]  J. Pierce,et al.  On spline functions determined by singular self-adjoint differential operators☆ , 1972 .

[24]  P. Laurent.  Approximation et optimisation , 1972 .

[25]  R. Schaback.  Konstruktion und algebraische Eigenschaften vonM-Spline-Interpolierenden , 1973 .

[26]  P. M. Prenter.  Splines and variational methods , 1975 .

[27]  Rui J. P. de Figueiredo.  LM-g splines☆ , 1975 .

[28]  Joseph W. Jerome,et al.  Minimum Norm Extremals in Function Spaces , 1975 .

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

[30]  Larry L. Schumaker,et al.  On pLg-splines , 1978 .

[31]  D. Elliott.  Lagrange interpolation — decline and fall?† , 1979 .

[32]  H. Weinert,et al.  Vector-valued Lg-splines I. Interpolating splines , 1979 .

[33]  Badri N. Sahney,et al.  The general problem of approximation and spline functions , 1979 .

[34]  Howard L. Weinert,et al.  Arma Splines, System Inverses, and Least-Squares Estimates , 1979 .

[35]  Paul L. Butzer,et al.  Functional Analysis and Approximation , 1981 .

[36]  G. Wahba.  Spline Interpolation and Smoothing on the Sphere , 1981 .

[37]  On nonlinear minimization problems and Lf-splines, I , 1983 .

[38]  Vimala Abraham.  On the existence and uniqueness of M-splines , 1985 .

[39]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[40]  G. Wahba.  Spline models for observational data , 1990 .

[41]  Guanrong Chen,et al.  PDL g splines defined by partial differential operators with initial and boundary value conditions , 1990 .

[42]  C. Rabut.  B-splines polyharmoniques cardinales : interpolation, quasi-interpolation, filtrage , 1990 .

[43]  L. Amodei,et al.  A vector spline approximation , 1991 .

[44]  Charles Swartz,et al.  An introduction to functional analysis , 1992 .

[45]  C. D. Boor,et al.  Multivariate piecewise polynomials , 1993, Acta Numerica.

[46]  $T_{f}$-splines et approximation par $T_{f}$ -prolongement , 1993 .

[47]  H. J. Taijeron,et al.  Spline Interpolation and Smoothing on Hyperspheres , 1994, SIAM J. Sci. Comput..

[48]  R. Franke,et al.  A Survey on Spherical Spline Approximation , 1995 .

[49]  Viacheslav I. Lebedev.  An introduction to functional analysis in computational mathematics , 1996 .

[50]  M. Hermite,et al.  Sur la formule d'interpolation de Lagrange. , .