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Strong uniqueness of generalized polynomial of best approximation having bounded coefficients

Strong uniqueness of generalized polynomial of best approximation having bounded coefficients Let the set of generalized polynomials haviug bounded coefficients be $$K = \left\{ {p = \sum\limits_{j = 1}^n {a_j g_j :a_j \in [a_j ,\beta _j ]} ,j = 1,2,...,n} \right\},$$ whereg 1,g 2, ...,g n are linearly independent continuous functions defined on the interval [a, b],α j ,β j are extended real numbers satisfyingα j <+∞,β j >−∞ andα j ⩽β j . Assume thatf is a continuous function defined on a compact setX ⊂ [a, b]. In the paper, we first give the sufficient conditions for the polynomial of best uniform approximation tof fromK being unique and strongly unique. Furthermore, we give two forms of necessary and sufficient conditions for the best approximation to be strongly unique. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Strong uniqueness of generalized polynomial of best approximation having bounded coefficients

Acta Mathematicae Applicatae Sinica , Volume 7 (1) – Jul 13, 2005

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Publisher
Springer Journals
Copyright
Copyright © 1991 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02080200
Publisher site
See Article on Publisher Site

Abstract

Let the set of generalized polynomials haviug bounded coefficients be $$K = \left\{ {p = \sum\limits_{j = 1}^n {a_j g_j :a_j \in [a_j ,\beta _j ]} ,j = 1,2,...,n} \right\},$$ whereg 1,g 2, ...,g n are linearly independent continuous functions defined on the interval [a, b],α j ,β j are extended real numbers satisfyingα j <+∞,β j >−∞ andα j ⩽β j . Assume thatf is a continuous function defined on a compact setX ⊂ [a, b]. In the paper, we first give the sufficient conditions for the polynomial of best uniform approximation tof fromK being unique and strongly unique. Furthermore, we give two forms of necessary and sufficient conditions for the best approximation to be strongly unique.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

References