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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
MathSciNet review: 1333314
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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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Additional Information

R. C. Gayle
Affiliation: Department of Science and Mathematics, Montana State University-Northern, P. O. Box 7751, Havre, Montana 59501

J. M. Wolfe
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

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