Unicity in piecewise polynomial $L^1$-approximation via an algorithm
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- by R. C. Gayle and J. M. Wolfe PDF
- Math. Comp. 65 (1996), 647-660 Request permission
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 [Uniqueness of optimal piecewise polynomial $L_{1}$ approximations for generalized convex functions, from âFunctional Analysis and Approximationâ, Internat. Ser. Numer. Math., vol. 60 (1981), 421â432]; 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 [J. Chow, On the uniqueness of best $L_{2}[0,1]$ approximation by piecewise polynomials with variable breakpoints, Math. Comp. 39 (1982), 571â585.]. 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 [Optimal segmented polynomial $L^{p}$-approximation, Computing 26 (1981), 239â246.]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the $L^{2}$ case.References
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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
- Received by editor(s): April 13, 1994
- Received by editor(s) in revised form: January 10, 1995
- © Copyright 1996 American Mathematical Society
- 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