Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-10T22:58:28.334Z Has data issue: false hasContentIssue false
  • English
  • Français

A variational approach to splines

Published online by Cambridge University Press:  17 February 2009

R. Champion ,
C. T. Lenard  and
T. M. Mills
R. Champion
Affiliation:
Division of Mathematics, La Trobe University, PO Box 199, Bendigo 3552, Australia.
C. T. Lenard
Affiliation:
Division of Mathematics, La Trobe University, PO Box 199, Bendigo 3552, Australia.
T. M. Mills
Affiliation:
Division of Mathematics, La Trobe University, PO Box 199, Bendigo 3552, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 1 , July 2000 , pp. 119 - 135
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Abraham, V., “On the existence and uniqueness of M-splines”, J. Approx. Theory 43 (1985) 36–42.CrossRefGoogle Scholar
[2]Ahlberg, J. H., Nilson, E. N. and Walsh, J. L., “Fundamental properties of generalized splines”, Proc. Nat. Acad. Sci. USA 52 (1964) 1412–1419.CrossRefGoogle ScholarPubMed
[3]Ahlberg, J. H., Nilson, E. N. and Walsh, J. L., The Theory of splines and Their Applications (Academic Press, New York, 1967).Google Scholar
[4]Amodei, L., “Étude d'une classe de fonctions splines vectorielles en vue de l'approximation d'un champ de vitesse. Applications à la météorologie”, Thèse, Université Paul Sabatier, Toulouse, 1993.Google Scholar
[5]Amodei, L. and Benbourhim, M. N., “A vector spline approximation”, J. Approx. Theory 67 (1991) 51–79.CrossRefGoogle Scholar
[6]Anselone, P. M. and Laurent, P. J., “A general method for the construction of interpolating or smoothing spline-functions”, Numer. Math. 12 (1968) 66–82.CrossRefGoogle Scholar
[7]Atteia, M., “Généralisation de la définition et des propriétés des spline fonctions”, C.R. Acad. Sc. Paris 260 (1965) 3550–3553.Google Scholar
[8]Atteia, M., “Étude de certains noyaux et théorie des fonctions spline en analyse numérique”, Thèse, I'Instit. Math. Appl., Grenoble, 1966.Google Scholar
[9]Atteia, M., Hilbertian Kernels and Spline Functions (North Holland, Amsterdam, 1992).Google Scholar
[10]Benbourhim, N. M. and Gaches, J., “Tf-splines et approximation par Tf-prolongement”, Studia Math. 106 (1993) 203–211.CrossRefGoogle Scholar
[11]Bezhaev, A. Yu. and Vasilenko, V. A., Variational Spline Theory (Novosibirsk Computing Center, Novosibirsk, 1993).Google Scholar
[12]Copley, P. and Schumaker, L. L., “On pLg-splines”, J. Approx. Theory 23 (1978) 1–28.CrossRefGoogle Scholar
[13]de Boor, C., “Best approximation properties of spline functions of odd degree”, J. Math. Mech. 12 (1963) 747–749.Google Scholar
[14]de Boor, C., “Multivariate piecewise polynomials”, Acta Numerica (1993) 65–109.CrossRefGoogle Scholar
[15]de Boor, C. and Lynch, R. E., “On splines and their minimum properties”, J. Math. Mech. 15 (1966) 953–969.Google Scholar
[16]de Figueiredo, R. J. P., “LM-g splines”, J. Approx. Theory 19 (1979) 332–360.CrossRefGoogle Scholar
[17]de Figueiredo, R. J. P. and Chen, G., “PDLg splines defined by partial differential operators with initial and boundary value conditions”, SIAM J. Numer. Anal. 27 (1990) 519–528.CrossRefGoogle Scholar
[18]Duchon, J., “Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces”, R.A.I.R.O. Analyse Numérique 10 (1976) 5–12.CrossRefGoogle Scholar
[19]Elliott, D., “Lagrange interpolation—decline and fall?”, Internal. J. Math. Ed. Sci. Tech. 10 (1979) 1–12.CrossRefGoogle Scholar
[20]Fisher, S. D. and Jerome, J. W., “Elliptic variational problems in L 2 and L ∞”, Indiana Univ. Math. Jnl. 23 (1974) 685–698.CrossRefGoogle Scholar
[21]Fisher, S. D. and Jerome, J. W., Minimum Norm Extremals in Function Spaces, Lecture Notes in Math. 479 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[22]Freeden, W., Schreiner, M. and Franke, R., “A survey of spherical spline approximation”, Surveys on Mathematics for Industry (to appear).Google Scholar
[23]Golomb, M., “Splines, nwidths and optimal approximations”, MRC Technical Summary Report 784, Mathematics Research Center, US Army, Madison, Wisconsin, 1967.Google Scholar
[24]Golomb, M. and Weinberger, H. F., “Optimal approximation and error bounds”, in On Numerical Approximation (ed. Langer, R. E.), (Univ. Wisconsin Press, Madison, 1959) 117–190.Google Scholar
[25]Greville, T. N. E., “Interpolation by generalized spline functions”, MRC Technical Summary Report 476, Mathematics Research Center, US Army, Madison, Wisconsin, 1964.Google Scholar
[26]Harder, R. L. and Desmarais, R. N., “Interpolation using surface splines”, J. Aircraft 9 (1972) 189–191.CrossRefGoogle Scholar
[27]Hartman, P., Ordinary Differential Equations, second ed. (Birkhaüser, Boston, 1982).Google Scholar
[28]Hermite, Ch., “Sur la formule d'interpolation de Lagrange”, J. Reine Angew. Math. 84 (1878) 70–79.CrossRefGoogle Scholar
[29]Holladay, J. C., “A smoothest curve approximation”, Math. Tables Aids Comput. 11 (1957) 233–243.CrossRefGoogle Scholar
[30]Holland, A. S. B. and Sahney, B. N., The General Problem of Approximation and Spline Functions (Robert E. Krieger Pub., Huntington, 1979).Google Scholar
[31]Holmes, R., “R-splines in Banach spaces: I. Interpolation of linear manifolds”, J. Math. Anal. Appl. 40 (1972) 574–593.CrossRefGoogle Scholar
[32]Jerome, J. W. and Pierce, J., “On spline functions determined by singular self-adjoint differential operators”, J. Approx. Theory 5 (1972) 15–40.CrossRefGoogle Scholar
[33]Jerome, J. W. and Schumaker, L. L., “On Lg-splines”, J. Approx. Theory 2 (1969) 29–49.CrossRefGoogle Scholar
[34]Jerome, J. W. and Varga, R. S., “Generalizations of spline functions and applications to nonlinear boundary value and eigenvalue problems”, in Theory and Applications of Spline Functions (ed. Greville, T. N. E.), (Academic Press, New York, 1969) 103–155.Google Scholar
[35]Lagrange, J.-L., “Memoire sur la méthode d'interpolation”, in Oeuvres de Lagrange (ed. J.-A., Serret), Volume 5, (Gauthiers-Villars, Paris, 1870) 663–684.Google Scholar
[36]Laurent, P.-J., Approximation et Optimisation (Hermann, Paris, 1972).Google Scholar
[37]Lebedev, V. I., An Introduction to Functional Analysis and Computational Mathematics (Birkhaüser, Boston, 1997).Google Scholar
[38]Lucas, T. R., “A theory of generalized splines with applications to nonlinear boundary value problems”, Ph. D. Thesis, Georgia Institute of Technology, 1970.Google Scholar
[39]Lucas, T. R., “M-splines”, J. Approx. Theory 5 (1972) 1–14.CrossRefGoogle Scholar
[40]Pai, D. V., “On nonlinear minimization problems and Lf -splines. I”, J. Approx. Theory 39 (1983) 228–235.CrossRefGoogle Scholar
[41]Powell, M. J. D., “The theory of radial basis function approximation”, in Adv. Num. Anal. Vol 2 (ed. Light, W.), (OUP, Oxford, 1992).Google Scholar
[42]Prenter, P. M., Splines and Variational Methods (John Wiley and Sons, New York, 1975).Google Scholar
[43]Rabut, C., “B-splines polyharmoniques cardinales: Interpolation, quasi-interpolation, filtrage”, Thèse, Université Paul Sabatier, Toulouse, 1990.Google Scholar
[44]Schaback, R., “Konstruktion und algebraische Eigenschaften von M-Spline-Interpolierenden”, Numer. Math. 21 (1973) 166–180.CrossRefGoogle Scholar
[45]Schoenberg, I. J., “Contributions to the problem of approximation of equidistant data by analytic functions. Parts A and B”, Quart. Appl. Math. 4 (1946) 45–99, 112–141.CrossRefGoogle Scholar
[46]Schoenberg, I. J., “On trigonometric spline interpolation”, J. Math. Mech. 13 (1964) 795–825.Google Scholar
[47]Schoenberg, I. J., “On the Ahlberg-Nilson extension of spline interpolation: The gsplines and their optimal properties”, J. Math. Anal. Appl. 21 (1968) 207–231.CrossRefGoogle Scholar
[48]Schultz, M. H. and Varga, R. S., “L-splines”, Numer. Math. 10 (1967) 345–369.CrossRefGoogle Scholar
[49]Sidhu, G. S. and Weinert, H. L., “Vector-valued Lg-splines. I. Interpolating splines”, J. Math. Anal. Appl. 70 (1979) 505–529.CrossRefGoogle Scholar
[50]Taijeron, H. J., Gibson, A. G. and Chandler, C., “Spline interpolation and smoothing on hyperspheres”, SIAM J. Sci. Comput. 15 (1994) 1111–1125.CrossRefGoogle Scholar
[51]Thomann, J., “Détermination et construction de fonctions Spline à deux variables définies sur un domaine rectangulaire ou circulaire”, Thèse, Université de Lille, 1970.Google Scholar
[52]Utreras, F. I., “Convergence rates for constrained spline functions”, Rev. Mat. Apl. 9 (1987) 87–95.Google Scholar
[53]Varga, R. S., Functional Analysis and Approximation Theory (SIAM, Philadelphia, 1971).CrossRefGoogle Scholar
[54]Wahba, G., “Spline interpolation and smoothing on the sphere”, SIAM J. Sci. Stat. Comp. 2 (1981) 5–16.CrossRefGoogle Scholar
[55]Wahba, G., Spline Models for Observational Data (SIAM, Philadelphia, 1990).CrossRefGoogle Scholar
[56]Weinert, H. L., Sesai, U. B. and Sidhu, G. S., “Arma splines, system inverses, and least-squares estimates”, SIAM J. Control Optim. 17 (1979) 525–536.CrossRefGoogle Scholar