Local strong uniqueness

Egger, A.; Taylor, G. D.

doi:10.1016/0021-9045(89)90028-2

Journal of Approximation Theory

1989

DOI: 10.1016/0021-9045(89)90028-2

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Local strong uniqueness

A. Egger

G. D. Taylor

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1994

1995

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Cited by 2 publications

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“…Two facts which demonstrate the importance of the concept of strong unicity were pointed out in [6]: Suppose /* and g* are (M,7)-strongly unique best approximants to / and g respectively.…”

mentioning

confidence: 90%

Strong unicity versus modulus of convexity

Huotari

Sahab

1994

Bull. Austral. Math. Soc.

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We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p-type modulus of convexity implies order p strong unicity of the simultaneous approximant.The study of strong unicity, initiated by Newman and Shapiro [9], resulted from a concern with the conditioning of the best approximation problem. In the present paper, the relation between strong unicity and modulus of convexity is explored. Throughout our discussion, we suppose that (X, ||-||) is a Banach space with modulus of convexity 8 > 0, and that V is a closed subspace of X. Bjornestal [2] showed that if x 0 G S(X) and if ||zo -2/|| ^ ll z o|| for every y G V, then, for every y G V with ||y|| ^ 3, that is, the vector 0 is a strongly unique best approximation to xo from V. We shall now show that Bjornestal's inequality characterises the modulus of convexity in a certain context, and that an analogue of this inequality holds in the non-uniformly convex space which underlies the theory of simultaneous approximation. We hope that these results will contribute to a deeper understanding of uniform convexity, as well as to the more "practical" goal of finding conditions under which tlie best approximation operator is continuous.We begin with precise definitions of the relevant concepts. Suppose that f £ X is fixed. An element z G V is said to be a best \\-\\-approximant to / from V if \\z -f\\ = inf{||w -f\\ : w G V}. Since X is uniformly convex, there is exactly one such z, and we denote it by /*. We say that /* is (M,j)-strongly unique of order p if, for some M -M(f) > 0, there exists a 7 = y(f,M) > 0 such that, for all v G V with | | » -/ ' | | < M

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“…Two facts which demonstrate the importance of the concept of strong unicity were pointed out in [6]: Suppose /* and g* are (M,7)-strongly unique best approximants to / and g respectively.…”

mentioning

confidence: 90%

Strong unicity versus modulus of convexity

Huotari

Sahab

1994

Bull. Austral. Math. Soc.

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show abstract

Simultaneous approximation from convex sets

Shi

Huotari

1995

Computers & Mathematics with Applications

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Local strong uniqueness

Cited by 2 publications

References 14 publications

Strong unicity versus modulus of convexity

Strong unicity versus modulus of convexity

Simultaneous approximation from convex sets

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