Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Characterizing shape preserving $ L\sb 1$-approximation


Author: D. Zwick
Journal: Proc. Amer. Math. Soc. 103 (1988), 1139-1146
MSC: Primary 41A15; Secondary 26A51
DOI: https://doi.org/10.1090/S0002-9939-1988-0954996-4
MathSciNet review: 954996
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: With certain restrictions, a characterization of the best $ {L_1}$-approximation to a continuous function from the set of $ n$-convex functions is proved. Under these restrictions the best approximation is shown to be unique. The case $ n = 2$ (convex functions) is considered in more detail.


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

  • [1] A. L. Brown, Best approximation by continuous $ n$-convex functions, J. Approximation Theory (to appear).
  • [2] A. S. Cavaretta, Jr., Oscillatory and zero properties for perfect splines and monosplines, J. Analyse Math. 28 (1975), 41-59.
  • [3] R. B. Darst and R. Huotari, Monotone $ {L_1}$-approximation on the unit $ n$-cube, Proc. Amer. Math. Soc. 95 (1985), 425-428. MR 806081 (87e:41008)
  • [4] R. Huotari, D. Legg and D. Townsend, Existence of best $ n$-convex approximants in $ {L_1}$, Preprint.
  • [5] -, Best $ {L_1}$-approximation by convex functions of several variables, Preprint.
  • [6] R. Huotari and A. D. Meyerowitz, Best $ {L_1}$ -approximation by convex functions, Unpublished manuscript.
  • [7] R. Huotari, A. D. Meyerowitz and M. Sheard, Best monotone approximants in $ {L_1}[0,1]$, J. Approximation Theory 47 (1986), 85-90. MR 843457 (87h:41030)
  • [8] R. Huotari and D. Zwick, Approximation in the mean by convex functions, Preprint. MR 1002887 (90f:41042)
  • [9] S. Karlin and W. J. Studden, Tchebyeheff systems: With applications in analysis and statistics, Wiley Interscience, New York, 1966. MR 0204922 (34:4757)
  • [10] C. A. Micchelli, Best $ {L_1}$-approximation by weak Chebyshev systems and the uniqueness of interpolating perfect splines, J. Approximation Theory 19 (1977), 1-14. MR 0454479 (56:12730)
  • [11] T. Popoviciu, Les fonctions convexes, Hermann, Paris, 1944. MR 0018705 (8:319a)
  • [12] -, Sur le prolongement des fonctions convexes d'ordre supérieur, Bull. Math. Soc. Roumaine des Sciences 36 (1934), 75-108.
  • [13] A. W. Roberts and D. E. Varberg, Convex functions, Academic Press, New York, 1973. MR 0442824 (56:1201)
  • [14] L. Schumaker, Spline functions: basic theory, Wiley, New York, 1981. MR 606200 (82j:41001)
  • [15] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin and New York, 1970. MR 0270044 (42:4937)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A15, 26A51

Retrieve articles in all journals with MSC: 41A15, 26A51


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0954996-4
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society