Abstract
The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pinkus, A., uniqueness in vector-valued approximation, J. Approx. Theory, 1993, 73: 17–92.
Li, C., Watson, G. A., On best simultaneous approximation, J. Approx. Theory, 1997, 91: 332–348.
Tanimoto, S., A characterization of best simultaneous approximations, J. Appmx. Theory, 1989, 59: 359–361.
Watson, G. A., A characterization of best simultaneous approximations, J. Approx. Theory, 1993, 75: 175–182.
Li, C., Watson, G. A., On best simultaneous appmximation of a finite set of functions, Computer Math. Applic., 1999, 37: 1–9.
Stechkin, S. B., Approximation properties of sets in norrned linear spaces, Rev. Roumaine Math. Pures and Appl. (in Russian), 1963, 8: 5–18.
Borwein, J. M., Fitzpatrick, S., Existence of nearest points in Banach spaces, Can. J. Math., 1989, 41: 702–720.
De Blasi, F.S., Myjak, J., On a generalized best appmximation problem, J. Approx. Theory, 1998, 94: 54–72.
Georgiev, P. G., The stmng Ekeland variational principle, the strung drop theorem and applications, J. Math. Anal. Appl., 1988, 131: 1–21.
Lau, K. S., Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J., 1978, 27: 791–795.
Li, C., On mutually nearest and mutually furthest points in reflexive Banach spaces, J. Approx. Theory, 2000, 103: 1–17.
Li, C., On well posed generalized best approximation problems, J. Approx. Thory, 2000, 107: 96–108.
Li, C, Almost K-Chebyshev sets, Acta. Math. Sin. (in Chinese), 1990, 33: 251–259.
De Blasi, F. S., Myjak, J., Papini, P. L., On mutually nearest and mutually furthest points of sets in Banach spaces, J. Appmx. Theory, 1992, 70: 142–155.
Dontchev, A., Zolezzi, T., Well Posed Optimization Problems, Lecture Notes in Math., Vol. 1543, New York: Springer- Verlag, 1993.
Li, C., Wang, X. H., Almost Chebyshev set with respect to bounded subsets, Science in China, Ser. A, 1997, 40: 375–383.
Ni, R. X., Li, C., On well posedness of farthest and simultaneous farthest problems in Banach spaces, Act. Math. Sinica (in Chinese), 2000, 43: 421–426.
De Blasi, F. S., Myjak. J., Ambiguous loci in best approximation theory, in Approximation Theory, Spline Functions and Application (ed. Singh, S. P.), Dordrecht: Kluwer Academic Publishers, 1992, 341–349.
De Blasi, F. S., Myjak, J., Ambiguous loci of the nearest point mapping in Banach spaces, Arch. Math., 1993, 61: 377–384.
De Blasi, F. S., Myjak, J., Papini, P. L., Pomus sets in best approximation theory, J. London Math. Sac., 1991, 44 (2): 135–142.
De Blasi, F. S., Kenderov, P. S., Myjak, J. et al., Ambiguous loci of the metric pmjection onto compact starshaped sets in a Banach space, Mh. Math., 1995, 119: 23–36.
Zhivkov, N. V.. Compacta with dense ambiguous loci of metric projections and antipmjections. Roc. Amer. Math. Soc., 1995, 123: 3403–3411.
Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math., New York: Springer-Verlag, 1989, 1364.
Diestel, J., Geometry of Banach Spaces-Selected Topic, Lect. Notes in Math.,New York: Springer-Verlag. 1975, 485.
Xu, S. Y., Li, C., Yang, W. S., Nonlinear Approximation Theory in Banach Spaces, Beijing: Science Press, 1998.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, C. On well posedness of best simultaneous approximation problems in Banach spaces. Sci. China Ser. A-Math. 44, 1558–1570 (2001). https://doi.org/10.1007/BF02880795
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02880795