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Unicity in piecewise polynomial $L^{1}$-approximation via an algorithm


Authors: R. C. Gayle and J. M. Wolfe
Journal: Math. Comp. 65 (1996), 647-660
MSC (1991): Primary 41A15, 41A52; Secondary 41A05
DOI: https://doi.org/10.1090/S0025-5718-96-00709-0
MathSciNet review: 1333314
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Abstract | References | Similar Articles | Additional Information

Abstract: Our main result shows that certain generalized convex functions on a real interval possess a unique best $L^{1}$ approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [11]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the $L^{2}$ case in [5]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [10]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the $L^{2}$ case.


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  • 1. J. Al-Siwan, Best segmented $L_{2}$-approximation with free knots, Dissertation, University of Oregon, Eugene, March 1986.
  • 2. D. L. Barrow, C. K. Chui, P. W. Smith, and J. D. Ward, Unicity of best mean approximation by second order splines with variable knots, Math. Comp. 32 (1978), 1132--1148. MR 58:1853
  • 3. D. Braess, ``Nonlinear Approximation Theory'', Springer-Verlag, Berlin-New York, 1986. MR 88e:41002
  • 4. H. G. Burchard, Piecewise polynomial approximation on optimal meshes, Journal of Approximation Theory 14 (1975), 128--147. MR 51:10957
  • 5. J. Chow, On the uniqueness of best $L_{2}[0,1]$ approximation by piecewise polynomials with variable breakpoints, Math. Comp. 39 (1982), 571--585. MR 83j:41009
  • 6. P. J. Davis, ``Interpolation and Approximation'', Blaisdell, New York, 1963. MR 28:393
  • 7. R. L. Eubank, On the relationship between functions with the same optimal knots in spline and piecewise polynomial approximation, Journal of Approximation Theory 40 (1984), 327--332. MR 85j:41016
  • 8. K. Jetter, $L_{1}$ Approximation verallgemeinerter konvexer Funktionen durch Splines mit freien Knoten, Math. Z. 164 (1978), 55--66. MR 80f:41007
  • 9. J. B. Kioustelidis, Optimal segmented approximations, Computing 24 (1980), 1-8. MR 82i:
    41032
  • 10. J. B. Kioustelidis, Optimal segmented polynomial $L^{p}$-approximation, Computing 26 (1981), 239--246. MR 82h:41006
  • 11. J. B. Kioustelidis, Uniqueness of optimal piecewise polynomial $L_{1}$ approximations for generalized convex functions, from ``Functional Analysis and Approximation'', Internat. Ser. Numer. Math., vol. 60 (1981), 421--432. MR 83d:41036
  • 12. J. M. Ortega, ``Numerical Analysis, a Second Course", Academic Press, New York, 1972. MR 53:6967
  • 13. A. M. Pinkus, ``On $L^{1}$-Approximation", Cambridge Univ. Press, Cambridge and New York, 1989. MR 90j:41046
  • 14. M. J. D. Powell, ``Approximation Theory and Methods'', Cambridge University Press, Cambridge, 1981. MR 82f:41001
  • 15. J. M. Wolfe, On the convergence of an algorithm for discrete $L_{p}$ approximation, Numer. Math. 32 (1979), 439--459. MR 80i:65021

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Additional Information

R. C. Gayle
Affiliation: Department of Science and Mathematics, Montana State University-Northern, P. O. Box 7751, Havre, Montana 59501
Email: gayle@nmc1.nmclites.edu

J. M. Wolfe
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: wolfe@bright.uoregon.edu

DOI: https://doi.org/10.1090/S0025-5718-96-00709-0
Keywords: Polynomial approximation, Lagrange interpolation, $L^{1}$ approximation
Received by editor(s): April 13, 1994
Received by editor(s) in revised form: January 10, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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