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Ahlberg, J. H., andE. N. Nilson: Convergence properties of the spline fit. J. Soc. Ind. Math.11, 95–104 (1963).
——, andJ. L. Walsh: Fundamental properties of generalized splines. Proc. Nat. Acad. Sci. (USA)52, 1412–1419 (1964).
—— Orthogonality properties of spline functions. J. Math. Analysis and applications11, 321–337 (1965)
——, andJ. L. Walsh: Convergence properties of generalized splines. Proc. Nat. Acad. Sci. (USA)54, 344–350 (1965).
——, andJ. L. Walsh Extremal, orthogonality and convergence properties of multidimensional splines. J. of Math. anal. and appl.12, 27–48 (1965).
— Best approximation and convergence properties of higher-order spline approximations. J. of Math. and Mech.14, No. 2, 231–244 (1965).
—— The approximation of linear functionals. J. SIAM, Num. Anal.3, No. 2, 173–182 (1966).
Ahlin, A. C.: Computer algorithms and theorems for generalized spline interpolation. SIAM National Meeting, N.Y., June 7–9, 1965
Atteia, M.: Generalisation de la définition et des propriétés des ≪ aspline-fonctions ≫. C. R. Acad. Sci. Paris260, 3550–3553 (1965).
——: Fonctions-spline généralisées. C. R. Acad. Sci. Paris261, 2149–2152 (1965).
——: Existence et détermination des fonctions spline à plusieurs variables. C. R. Acad. Sci. Paris262, 575–578 (1966).
— Théorie et applications des fonctions-spline en analyse numérique. Thèse, Grenoble (1966).
— Sur les fonctions-spline généralisées. Sème Congrès de l'AFIRO, Lille 27 juin -1 er juillet 1966.
Birkhoff, G., andH. L. Garabedian: Smooth surface interpolation. J. Math. and Physics39, 258–268 (1960).
——, andC. De Boor: Error bounds for spline interpolation. J. of Math. and Mech.13, No. 5, 827–835 (Sept. 1964).
——: Piecewise polynomial interpolation and approximation. Approximation of functions,H. L. Garabedian (ed.), pp. 164–190. Amsterdam: Elsevier 1965.
Carasso, C.: Méthodes numériques pour l'obtention de fonctions-spline. Thèse de 3éme Cycle, Université de Grenoble, 28 mars 1966.
— Construction numérique de fonctions-spline. Vème Congrès de l'AFIRO, Lille 27 juin- 1 er juillet 1966.
—— Méthode générale de construction de fonctions-spline. Revue française d'informatique et de Recherche opérationnelle5, 119–127 (1967).
Curry, H. B., andI. J. Schoenberg: On Polya frequency functions IV: The spline functions and their limits. Bull. Amer. Math. Soc.53, 1114 (1947)
Boor, C. de: Bicubic spline interpolation. J. Math. Phys.41, 212–218 (1962).
——: Best approximation properties of spline functions of odd degree. J. Math. Mech.12, 747–750 (1963).
——, andR. E. Lynch: On splines and their minimum properties. J. Math. Mech.15, 953–969 (1966).
Golomb, M., andH. Weinberger: Optimal approximation and error bounds. In “On numerical Approximation”,R. E. Langer (ed.), pp. 117–190. Madison: The Univ. of Wisconsin Press 1959.
— Lectures on theory of approximation. Argonne National Laboratory. Appl. Math. Division (1962).
Greville, T. N. E.: The general theory of osculatory interpolation. Trans. of the Acturial Society of America45, 202–265 (1944).
——: Subtabulaçaô por minimas quadrados de diferenças finitas. Bol. Inst. Brasil. Atuaria2, 7–34 (1946).
——, andH. Vaughan: Polynomial interpolation in terms of symbolic operators. Trans. Soc. Actuar.6, 413–476 (1954).
——: Interpolation by generalized spline functions. SIAM Review6, 483 (1964).
—: Numerical procedures for interpolation by spline functions. Math. Res. Center, United States Army, The Univ. of Wisconsin, Contract No. DA-11-022-ORD-2059. MRC Techn. Summary report, 450, january 1964. J. SIAM, Num. Anal. 1, 53–68 (1964).
—, andI. J. Schoenberg: Smoothing by generalized spline-functions.
Johnson, R. S.: On monosplines of least deviation. Trans. Amer. Math. Soc.96, 458–477 (1960).
Joly, J. L.: Utilisation des Fonctions-spline pour le lissage. Vème Congrès de l'AFIRO, Lille, 27 juin-ler juillet 1966.
— Convergence des fonctions-spline (à paraître).
——: Théorèmes de convergence pour les fonctions-spline générales d'interpolation et d'ajustement. C. R. Acad. Sei. Paris264, Ser. A, 126–128 (1967).
Laurent, P. J.: Propriétés des fonctions-spline et meilleure approximation au sens de SARD. Cycle de conférences de la chaireJ. von Neumann, 1965/66, Université libre de Bruxelles.
— Théorèmes de caractérisation en approximation convexe. Colloque sur la théorie de l'approximation des fonctions. Cluj (Roumanie) - 15–20 septembre 1967. Mathematica 30 (33), 1, 95–111 (1968).
—: Représentation de données expérimentales à l'aide de fonctions-spline d'ajustement et évaluation optimale de fonctionnelles et évaluation optimale de fonctionnelles linéaires continues. Colloque: Problèmes fondamentaux de calcul numérique Prague, 11–15 septembre 1967. Aplikace Matematiky13, 154–162 (1968).
Reinsch, Ch.: Smoothing by Spline Functions. Num. Math.10, 177–183 (1967).
Sard, A.: Linear approximation. American Mathematical Society (1963).
Schoenberg, I. J.: Contributions to the problem of approximation of equidistant data by analytic functions. Part A. Quart. Appl. Math.4, 45–99 (1946).
——, etU.A. Whitney: Sur la positivité des déterminants de translations des fonctions de fréquence de Polya avec une application au problème d'interpolation par les fonctions ≪ spline ≫. C. R. Acad. Sei. Paris228, 1996–1998 (1949).
——: On Polya frequency functions. III. The positivity of translation determinants with on application to the interpolation problem by spline curves. Trans. Amer. Math. Soc.74, 246–259 (1953)
——: Spline functions, convex curves, and mechanical quadrature. Bull. Amer. Math. Soc.64, 352–257 (1958).
— On interpolation by spline functions and its minimal properties. Proc. of the Conference on Approximation theory, Oberwolfach, Germany, August 1963.
— Address given at SIAM, Conference on approximation. Gatlinburg, Tennessee, October 24 (1963).
— On best approximation of linear operators. Kon. Nederlandse Akad, van Wetenschappen, Proceedings, Series A,67, 155–163 (1964).
——: On trigonometric spline interpolation. J. of Math. and Mech.13, No. 5, 795–825 (Sept 1964).
——: Spline interpolation and best quadrature formulae. Bull Amer. Math. Soc.70, No. 1, 143–148 (1964).
——: Spline functions and the problem of graduation. Proc. Nat. Acad. Sci.52, 947–950 (1964).
——: Spline interpolation and the higher derivatives. Proc. of the Nat. Acad. Sci.51, No. 1, 24–28 (1964).
—, andT. N. E. Greville: Smoothing by generalized spline-functions. SIAM National Meeting, N.Y., June 7-9, 1965 (Preprints).
——: On monosplines of least deviation and best quadrature formulae. J. SIAM. Anal.2, 144–170 (1965).
——: On monosplines of least square deviation and best quadrature formulae II. J. SIAM, Num. Anal.3, No. 2, 321–328 (1966).
Walsh, J. L., J. H. Ahlberg, andE. N. Nilson: Best approximation properties of the spline fit. J. Math. Mech.11, 225–234 (1962).
—— —— ——: Best approximation and convergence properties of higher-order spline fits. Amer. Math. Soc. Notices10, 202 (1963).
Weinberger, H. F.: Optimal approximation for functions prescribed at equally spaced points. J. of res. of the N.B.S.65 B, No. 2, 99–104 (1961).
Yosida, K.: Functional analysis. Berlin-Heidelberg-New York: Springer 1965.
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Anselone, P.M., Laurent, P.J. A general method for the construction of interpolating or smoothing spline-functions. Numer. Math. 12, 66–82 (1968). https://doi.org/10.1007/BF02170998
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DOI: https://doi.org/10.1007/BF02170998