A topological approach to Best Approximation Theory
DOI:
https://doi.org/10.4995/agt.2004.1994Keywords:
Wrapping, Best Approximation, Proximinality, CompactnessAbstract
The main goal of this paper is to put some light in several arguments that have been used through the time in many contexts of Best Approximation Theory to produce proximinality results. In all these works, the main idea was to prove that the sets we are considering have certain properties which are very near to the compactness in the usual sense. In the paper we introduce a concept (the wrapping) that allow us to unify all these results in a whole theory, where certain ideas from Topology are essential. Moreover, we do not only cover many of the known classical results but also prove some new results. Hence we prove that exists a strong interaction between General Topology and Best Approximation Theory.
Downloads
References
J. M. Almira, A. J. López-Moreno, N. Del Toro, Metrics with good corona properties, Questions and Answers in General Topology 21 (2003), 19-26.
F. S. De Blasi, J. Myjak, On a Generalized Best Approximation Problem, J. Approx. Theory 94 (1998), 54-72. http://dx.doi.org/10.1006/jath.1998.3177
P. L. Chebyshev, Théorie des mécanismes connus sous le nom de parallélogrammes, Mem. Acad. Sci. Petersb. 7 (1854), 539-568. Also to be found in Oeuvres de P. L. Tchebychef, Volume 1, 111-143, Chelsea, New York, 1961.
P. L. Chebyshev, Sur les questions de minima qui se rattachent Ó la représentation approximative des fonctions, Mem. Acad. Sci. Petersb. 7 (1859), 199-291. Also to be found in Oeuvres de P. L. Tchebychef, Volume 1, 273-378, Chelsea, New York, 1961.
M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313-317. http://dx.doi.org/10.1090/S0002-9904-1941-07451-3
F. Deutsch, Existence of Best Approximations, J. Approximation Theory 28 (1980), 132- 154. http://dx.doi.org/10.1016/0021-9045(80)90085-4
J. Dujundji, Topology, Allyn and Bacon, 1972.
N. Dunford, J. T. Schwartz, Linear Operators. Part I, John Wiley, 1988.
W. F. Eberlein, Weak compactness in Banach spaces I, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 51-53. http://dx.doi.org/10.1073/pnas.33.3.51
R. A. Hirschfeld, On best approximation in normed vector spaces, Nieuw Archief voor Wiskunde 6 (1958), 41-51.
V. L. Klee, Some characterizations of compactness, Amer. Math. Monthly 58 (1951), 389-393. http://dx.doi.org/10.2307/2306551
Yu. F. Korobeinik, On fixed points of one class of operators, Matematychni Studii 7 (1997), 187-192.
G. Köthe, Topological Vector Spaces I, Springer–Verlag, 1969.
S. Mazur, Über konvexe Mengen in linearen normierte Räumen, Studia Math. 4 (1933), 70-84.
E. Michael, Selection theorems with and without dimensional restrictions, in Recent Developments of General Topology and its Applications (Berlin, 1992) pp. 218-222, Akademie-Verlag, 1992.
C. Mustata, On the best approximation in metric spaces, Rev. Anal. Numer. Theor. Approx. 4 (1975), 45-50.
D. V. Pai, P. Govindarajulu, On Set–Valued f-Projections and f-Farthest Point Mappings, J. Approx. Theory 42 (1984), 4-13. http://dx.doi.org/10.1016/0021-9045(84)90050-9
R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238-255.
R. R. Phelps, Convex Sets and Nearest Points, Proc. Amer. Math. Soc. 8 (1957), 790-797. http://dx.doi.org/10.1090/S0002-9939-1957-0087897-7
F. Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1918), 71–98. http://dx.doi.org/10.1007/BF02422940
S. Romagera, M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory 103 (2000), 292-301. http://dx.doi.org/10.1006/jath.1999.3439
Samuel G. Moreno, F. Zó, Mejor Aproximación en Espacios Topológicos, Métricos y Vectoriales Topológicos, Magister Thesis Disertation, Universidad Nacional de San Luis, Argentina, 1997.
V. L. Smulian, About the principle of inclusion in spaces of type B , (in russian) Mat. Sbornik N. S. 5 (1939), 317-328. Traduced survey in Math. Rev. 1 (1940), 335.
I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer–Verlag, 1970.
A. F. Timan, Theory of approximation of functions of a real variable, Dover Publications (1994).
Downloads
How to Cite
Issue
Section
License
This journal is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike- 4.0 International License.