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L-splines. (English) Zbl 0183.44402

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[1] Ahlberg, J. H., andE. N. Nilson: Convergence properties of the spline fit. SIAM J. Appl. Math.11, 95–104 (1963). · Zbl 0196.48701 · doi:10.1137/0111007
[2] —- Orthogonality properties of spline functions. J. Math. Anal. Appl.11, 321–337 (1965). · Zbl 0136.04801 · doi:10.1016/0022-247X(65)90089-2
[3] —- The approximation of linear functionals. SIAM J. Numer. Anal.3, 173–182 (1966). · Zbl 0147.05102 · doi:10.1137/0703013
[4] —-, andJ. L. Walsh: Higher-order spline interpolation. Notices Amer. Math. Soc.11, 767 (1964) (abstract).
[5] —— Fundamental properties of generalized splines. Proc. Nat. Acad. Sci. U.S.A.52, 1412–1419 (1964). · Zbl 0136.36204 · doi:10.1073/pnas.52.6.1412
[6] —— Convergence properties of generalized splines. Proc. Nat. Acad. Sci. U.S.A.54, 344–350 (1965). · Zbl 0136.36301 · doi:10.1073/pnas.54.2.344
[7] —— Best approximation and convergence properties for higher order spline approximations. J. Math. Mech.14, 231–244 (1965). · Zbl 0141.06801
[8] —— Extremal, orthogonality, and convergence properties of multidimensional splines. J. Math. Anal. Appl.12, 27–48 (1965). · Zbl 0136.04802 · doi:10.1016/0022-247X(65)90051-X
[9] —— The theory of splines and their applications. New York: Academic Press 1967. · Zbl 0158.15901
[10] Birkhoff, G., H. Burchard, andD. Thomas: Nonlinear interpolation by splines, pseudo-splines, and elastica. Research Publication General Motors Corporation GMR 468 (1965).
[11] – andC. De Boor: Error bounds for spline interpolation. J. Math. Mech.13, 827–836 (1964). · Zbl 0143.28503
[12] —- Piecewise polynomial interpolation and approximation. Approximation of Functions,H. L. Garabedian (ed.) (pp. 164–190). Amsterdam: Elsevier Publishing Company 1965.
[13] —-B. Swartz, andB. Wendroff: Rayleigh-Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal.3, 188–203 (1966). · Zbl 0143.38002 · doi:10.1137/0703015
[14] – andH. L. Garabedian: Smooth surface interpolation. J. Math. and Phys.39, 258–268 (1960).
[15] Boor, C. De: Bicubic spline interpolation. J. Math. and Phys.41, 212–218 (1962). · Zbl 0108.27103
[16] – Best approximation properties of spline functions of odd degree. J. Math. Mech.12, 747–749 (1963). · Zbl 0116.27601
[17] –: On splines and their minimum properties. J. Math. Mech.15, 953–969 (1966). · Zbl 0185.20501
[18] Ciarlet, P. G., M. H. Schultz, andR. S. Varga: Numerical methods of highorder accuracy for nonlinear boundary value problems. I. One dimensional problem. Numer. Math.9, 394–430 (1967). · Zbl 0155.20403 · doi:10.1007/BF02162155
[19] Curry, H. B., andI. J. Schoenberg: On Pólya frequency functions IV. The fundamental spline functions and their limits. J. Analyse Math.17, 71–107 (1966). (MRC Tech. Summary Report 567 (1965).) · Zbl 0146.08404 · doi:10.1007/BF02788653
[20] Favard, J.: Sur l’interpolation. J. Math. Pures Appl. (9)19, 281–306 (1940). · Zbl 0026.01201
[21] Fowler, A. H., and C. W. Wilson: Cubic spline, a curve fitting routine. Report Y-1400, Oak Ridge (1963).
[22] Golomb, M., andH. F. Weinberger: Optimal approximation and error bounds. Proc. Symp. on Numerical Approximation,R. E. Langer (ed.) (pp. 117–190). Madison: Univ. of Wisconsin Press 1959
[23] Greville, T. N. E.: Numerical procedures for interpolation by spline functions. SIAM J. Numer. Anal.1, 53–68 (1965). (MRC Tech. Summary Report 450 (1964).) · Zbl 0141.33602
[24] – Interpolation by generalized spline functions. MRC Tech. Summary Report 476 (1964). · Zbl 0147.32101
[25] – Spline functions, interpolation, and numerical quadrature. Mathematical Methods for Digital Computers, Volume 2,A. Ralston andH. S. Wilf (eds.) (pp. 156–168). New York: John Wiley and Sons 1967.
[26] Holladay, J. C.: Smoothest curve approximation. Math. Tables Aids to Comp.11, 233–243 (1957). · Zbl 0084.34904 · doi:10.2307/2001941
[27] Johnson, R. S.: On monosplines of least deviation. Trans. Amer. Math. Soc.96, 458–477 (1960). · Zbl 0094.03903 · doi:10.1090/S0002-9947-1960-0122938-4
[28] Karlin, S., andZ. Ziegler: Tchebycheffian spline functions. SIAM J. Numer. Anal.3, 514–543 (1966). · Zbl 0171.31002 · doi:10.1137/0703044
[29] –, andW. J. Studden: Tchebycheff Systems: With Applications in Analysis and Statistics. New York: Interscience 1966. · Zbl 0153.38902
[30] Loscalzo, F. R., andT. D. Talbot: Spline function approximations for solutions of ordinary differential equations (6 pp.). MMS. from Computer Sciences Dept., Univ. of Wisconsin 1966. · Zbl 0171.36301
[31] MacLaren, D. H.: Formulas for fitting a spline curve through a set of points. Boeing Appl. Math. Report 2 (1958).
[32] Marsden, M., andI. J. Schoenberg: On variation diminishing spline approximation methods (27 pp.). MRC Tech. Summary Report 694 (1966). · Zbl 0171.31001
[33] Meyers, L. F., andA. Sard: Best interpolation formulas. J. Math. and Phys.29, 198–206 (1950). · Zbl 0040.02801
[34] Sard, A.: Best approximate integration formulas; best approximation formulas. Amer. J. Math.71, 80–91 (1949). · Zbl 0039.34104 · doi:10.2307/2372095
[35] – Linear Approximation, Math. Surveys No. 9. Providence, R. I.: Amer, Math. Soc. 1963.
[36] Schoenberg, I. J.: Contributions to the problem of approximation of equidistant data by analytic functions. Parts A and B. Quart. Appl. Math.4, 45–99, 112–141 (1946). · Zbl 0061.28804
[37] – On Pólya frequency functions I. The totally positive functions and their Laplace transforms. J. Analyse Math.1, 351–374 (1951). · Zbl 0045.37602 · doi:10.1007/BF02790092
[38] – Spline functions, convex curves and mechanical quadrature. Bull. Amer. Math. Soc.64, 352–357 (1958) · Zbl 0085.33701 · doi:10.1090/S0002-9904-1958-10227-X
[39] – On best approximations of linear operators. Nederl. Akad. Wetensch. Ser. A67, 155–163 (1964) and Indag. Math.26, 155–163 (1964). · Zbl 0146.08501
[40] – Spline interpolation and best quadrature formulae. Bull. Amer. Math. Soc.70, 143–148 (1964). · Zbl 0136.36202 · doi:10.1090/S0002-9904-1964-11054-5
[41] – Spline interpolation and the higher derivatives. Proc. Nat. Acad. Sci. U.S.A.51, 24–28 (1964). · Zbl 0136.36201 · doi:10.1073/pnas.51.1.24
[42] – On trigonometric spline interpolation. J. Math. Mech.13, 795–825 (1964). · Zbl 0147.32104
[43] – On monosplines of least deviation and best quadrature formulae. SIAM J. Numer. Anal. Ser. B2, 144–170 (1965). · Zbl 0136.36203 · doi:10.1137/0702012
[44] – On interpolation by spline functions and its minimal properties. Proc. Conf. on Approximation Theory in, Germany 1963 (pp. 109–129). Basel: Birkhauser 1964.
[45] – On monosplines of least square deviation and best quadrature formulae II. SIAM J. Numer. Anal.3, 321–328 (1966). · Zbl 0147.32103 · doi:10.1137/0703025
[46] – On spline functions (52 pp.). MRC Tech. Summary Report 625 (1966).
[47] – On Hermite-Birkhoff interpolation (9 pp.). MRC Tech. Summary Report 659 (May 1966).
[48] –On the Ahlberg-Nilson extension of spline interpolation: theg-splines and their optimal properties (35 pp.). MRC Tech. Summary Report 716 (1966).
[49] Schoenberg, I. J., andAnne Whitney: Sur la positivité des déterminants de translation des functions de fréquence de Pólya, aver une application à un problèm d’interpolation. C. R. Acad. Sci. Paris228, 1996–1998 (1949). · Zbl 0036.03302
[50] —- On Pólya frequency functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Amer. Math. Soc.74, 246–259 (1953). · Zbl 0051.33606
[51] Schweikert, D. G.: The spline in tension (hyperbolic spline) and the reduction of extraneous inflection points (82 pp.). Ph. D. Thesis, Brown University (1966). · Zbl 0146.14102
[52] Secrest, D.: Error bounds for interpolation and differentiation by the use of spline functions. SIAM J. Numer. Anal.2, 440–447 (1965). · Zbl 0143.38804
[53] Sharma, A., andA. Meir: Degree of approximation of spline interpolation. J. Math. Mech.15, 759–767 (1966). · Zbl 0158.30702
[54] Thacher, H., ed.: Numerical properties of functions of more than one independent variable. Ann. New York Acad. Sci.86, 677–874 (1960).
[55] Walsh, J. L., J. H. Ahlberg, andE. N. Wilson: Best approximation properties of the spline fit. J. Math. Mech.11, 225–234 (1962). · Zbl 0196.48603
[56] —— Best approximation and convergence properties of higher order spline fits. Notices Amer. Math. Soc.10, 202 (1963) (abstract).
[57] Weinberger, H. F.: Optimal approximation for functions prescribed at equally spaced points. J. Res. Nat. Bur. Standards Sect. B65 B, 99–104 (1961). · Zbl 0168.14901
[58] Wendroff, B.: Bounds for eigenvalues of some differential operators by the Rayleigh-Ritz method. Math. Comp.19, 218–224 (1965). · Zbl 0139.10703 · doi:10.1090/S0025-5718-1965-0179932-5
[59] Whittaker, E. T.: On the functions which are represented by the expansions of the interpolation-theory. Proc. Roy. Soc. Edinburgh35, 181–194 (1915). · JFM 45.0553.02
[60] Coddington, E. A., andN. Levinson: Theory of ordinary differential equations. New York: McGraw-Hill Book Co. 1955. · Zbl 0064.33002
[61] Hardy, G. H., J. E. Littlewood, andG. Pólya: Inequalities. Cambridge: Cambridge University Press 1952. · Zbl 0047.05302
[62] Hartman, P.: Ordinary differential equations. New York: John Wiley and Sons, Inc. 1964. · Zbl 0125.32102
[63] Opial, Z.: On a theorem of O. Aramă. J. Differential Equations.3, 88–91 (1967. · Zbl 0152.28103 · doi:10.1016/0022-0396(67)90008-3
[64] Pólya, G.: On the mean value theorem corresponding to a given linear homogeneous differential equation. Trans. Amer. Math. Soc.24, 233–243 (1922).
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