Unicity in piecewise polynomial -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 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 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 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

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