Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Characterisation theorem for best polynomial spline approximation with free knots


Authors: Nadezda Sukhorukova and Julien Ugon
Journal: Trans. Amer. Math. Soc. 369 (2017), 6389-6405
MSC (2010): Primary 49J52, 90C26, 41A15, 41A50
DOI: https://doi.org/10.1090/tran/6863
Published electronically: March 17, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterisation theorem for best free-knots polynomial spline approximation, which is stronger than the existing characterisation results, at least in the case when only continuity is required.


References [Enhancements On Off] (What's this?)

  • [1] Vladimir F. Demyanov and Alexander M. Rubinov, Constructive nonsmooth analysis, Approximation & Optimization, vol. 7, Peter Lang, Frankfurt am Main, 1995. MR 1325923
  • [2] Vladimir Demyanov and Alexander Rubinov (eds.), Quasidifferentiability and related topics, Nonconvex Optimization and its Applications, vol. 43, Kluwer Academic Publishers, Dordrecht, 2000. MR 1766790
  • [3] B. Luderer, On the quasidifferential of a continual maximum function, Optimization 17 (1986), no. 4, 447-452. MR 843035, https://doi.org/10.1080/02331938608843155
  • [4] G. Meinardus, G. Nürnberger, M. Sommer, and H. Strauss, Algorithms for piecewise polynomials and splines with free knots, Math. Comp. 53 (1989), no. 187, 235-247. MR 969492, https://doi.org/10.2307/2008358
  • [5] Bernd Mulansky, Chebyshev approximation by spline functions with free knots, IMA J. Numer. Anal. 12 (1992), no. 1, 95-105. MR 1152687, https://doi.org/10.1093/imanum/12.1.95
  • [6] Günther Nürnberger, Approximation by spline functions, Springer-Verlag, Berlin, 1989. MR 1022194
  • [7] G. Nürnberger, Bivariate segment approximation and free knot splines: Research Problems 96-4, Constr. Approx. 12 (1996), no. 4, 555-558. MR 1412199, https://doi.org/10.1007/BF02437508
  • [8] G. Nürnberger, L. L. Schumaker, M. Sommer, and H. Strauss, Approximation by generalized splines, J. Math. Anal. Appl. 108 (1985), no. 2, 466-494. MR 793660, https://doi.org/10.1016/0022-247X(85)90039-3
  • [9] G. Nürnberger, L. Schumaker, M. Sommer, and H. Strauss, Uniform approximation by generalized splines with free knots, J. Approx. Theory 59 (1989), no. 2, 150-169. MR 1022115, https://doi.org/10.1016/0021-9045(89)90150-0
  • [10] John R. Rice, Characterization of Chebyshev approximations by splines, SIAM J. Numer. Anal. 4 (1967), 557-565. MR 0223804
  • [11] Larry Schumaker, Uniform approximation by Chebyshev spline functions. II. Free knots, SIAM J. Numer. Anal. 5 (1968), 647-656. MR 0241867
  • [12] Nadezda Sukhorukova, Uniform approximation by the highest defect continuous polynomial splines: necessary and sufficient optimality conditions and their generalisations, J. Optim. Theory Appl. 147 (2010), no. 2, 378-394. MR 2726892, https://doi.org/10.1007/s10957-010-9715-0
  • [13] Nadezda Sukhorukova, Vallée Poussin theorem and Remez algorithm in the case of generalised degree polynomial spline approximation, Pac. J. Optim. 6 (2010), no. 1, 103-114. MR 2604519
  • [14] M. G. Tarashnin, Application of the theory of quasidifferentials to solving approximation problems, Ph.D. thesis, St-Petersburg State University, 1996, 119 pp. (in Russian).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 49J52, 90C26, 41A15, 41A50

Retrieve articles in all journals with MSC (2010): 49J52, 90C26, 41A15, 41A50


Additional Information

Nadezda Sukhorukova
Affiliation: Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia
Address at time of publication: Faculty of Science, Federation University, P.O. Box 663, Ballarat, Victoria 3353, Australia
Email: nsukhorukova@swin.edu.au

Julien Ugon
Affiliation: Centre for Informatics and Applied Optimization, Federation University, P.O. Box 663, Ballarat, Victoria 3353, Australia
Email: j.ugon@federation.edu.au

DOI: https://doi.org/10.1090/tran/6863
Received by editor(s): September 17, 2013
Received by editor(s) in revised form: September 18, 2013, and September 27, 2015
Published electronically: March 17, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society