Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T11:04:02.344Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong unicity versus modulus of convexity

Published online by Cambridge University Press:  17 April 2009

Robert Huotari  and
Salem Sahab
Robert Huotari
Affiliation:
Department of Mathematics IdahoState UniversityPocatello ID 83209United States of America
Salem Sahab
Affiliation:
Mathematics Department PO Box 9028King Abdulaziz UniversityJeddah 21413Saudi Arabia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 2 , April 1994 , pp. 305 - 310
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Amir, D. and Ziegler, Z., ‘Relative Chebyshev centers in normed linear spaces, I’, J. Approx. Theory 29 (1980), 235–252.CrossRefGoogle Scholar
[2]Bjornestal, B.O., ‘Local Lipschitz continuity of the metric projection operator’, Approx. Theory, Banach Center Publ, Vol. 4, PWN-Poliah Sci. Publ. (1979), 43–53.Google Scholar
[3]Day, M. M., Normed linear spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[4]Diestel, J., Geometry of Banach spaces – selected topics, Lecture Notes in Mathematics 485 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[5]Diestel, J., Sequences and series in Banach spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1984).CrossRefGoogle Scholar
[6]Egger, A. and Taylor, G.D., ‘Local strong uniqueness’, J. Approx. Theory 58 (1989), 267–280.CrossRefGoogle Scholar
[7]Goel, D.S., Holland, A.S.B., Nasim, C. and Sahney, B.N., ‘On best simultaneous approximation in normed linear spaces’, Canad. Math. Bull. 17 (1974), 523–527.CrossRefGoogle Scholar
[8]Lin, P.K., ‘Strongly unique best approximation in uniformly convex Banach spaces’, J. Approx. Theory 56 (1989), 101–107.CrossRefGoogle Scholar
[9]Newman, D.J. and Shapiro, H.S., ‘Some theorems on Chebyshev approximation theory’, Duke Math. J. 30 (1963), 681–683.CrossRefGoogle Scholar