Characterization of \(\varepsilon\)-nearest points in spaces with asymmetric seminorm

Costică Mustăţa

doi:10.33993/jnaat332-777

Authors

Costică Mustăţa Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat332-777

Keywords:

asymmetric seminormed spaces, \(\varepsilon\)-nearest points, characterization

Abstract views: 265

Abstract

In this note we are concerned with the characterization of the elements of \(\varepsilon\)-best approximation (\(\varepsilon\)-nearest points) in a subspace \(Y\) of space \(X\) with asymmetric seminorm. For this we use functionals in the asymmetric dual \(X^{b}\) defined and studied in some recent papers [1], [2], [5].

Downloads

Download data is not yet available.

References

Borodin, P. A., The Banach-Mazur theorem for spaces with an asymmetric norm and its applications in convex analysis, Mat. Zametki, 69, no. 3, pp. 193-217, 2001.

De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx. Theory, 94, no. 1, pp. 54-72, 1998, https://doi.org/10.1006/jath.1998.3177 DOI: https://doi.org/10.1006/jath.1998.3177

Cobzas, S. and Mustata, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 39-50, 2004, http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art5

Cobzas, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27, pp. 1-22, (275-296), 2004, https://doi.org/10.2989/16073600409486100 DOI: https://doi.org/10.2989/16073600409486100

Garcia-Raffi, L. M., Romaguera S. and Sánchez-Pérez, E. A., The dual space of an asymmetric normed linear space, Quaest. Math., 26, no. 1, pp. 83-96, 2003, https://doi.org/10.2989/16073600309486046 DOI: https://doi.org/10.2989/16073600309486046

Garcia-Raffi, L.M., Romaguera S. and Sánchez-Pérez, E. A., On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29, no. 3, pp. 717-728, 2003 (electronic).

Krein, M. G. and Nudel'man, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow 1973 (in Russian). English translation: American Mathematical Society, Providence, R.I., 1997.

Li, Chong and Ni, Renxing, Derivatives of generalized distance functions and existence of generalized nearest points, J. Approx. Theory, 115, no. 1, pp. 44-55, 2002, https://doi.org/10.1006/jath.2001.3651 DOI: https://doi.org/10.1006/jath.2001.3651

Mabizela, S., Characterization of best approximation in metric linear spaces, Scientiae Mathematicae Japonica, 57, 2, pp. 233-240, 2003. DOI: https://doi.org/10.1007/BF02835236

Mustăţa, C., On the best approximation in metric spaces, Mathematica -- Revue d'Analyse Numérique et de Théorie de l'Approximation, L'Analyse Numérique et la Théorie de l'Approximation, 4, pp. 45-50, 1975, http://ictp.acad.ro/jnaat/journal/article/view/1975-vol4-no1-art5

Mustăţa, C., On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Numér. Théor. Approx., 32, no. 2, pp. 187-192, 2003, http://ictp.acad.ro/jnaat/journal/article/view/2003-vol32-no2-art7

Ni, Renxing, Existence of generalized nearest points, Taiwanese J. Math., 7, no. 1, pp. 115-128, 2003, https://doi.org/10.11650/twjm/1500407521 DOI: https://doi.org/10.11650/twjm/1500407521

Pantelidis, G., Approximations theorie für metrich linear Räume, Math. Ann., 184, pp. 30-48, 1969, https://doi.org/10.1007/bf01350613 DOI: https://doi.org/10.1007/BF01350613

Rezapour, Sh., ε-pseudo Chebyshev and ε-quasi Chebyshev subspaces of Banach spaces, Technical Report, Azarbaidjan University of Tarbiot Moallem, 2003.

Rezapour, Sh., ε-weakly Chebyshev subspaces of Banach spaces, Analysis in Theory and Applications, 19, no. 2, pp. 130-135, 2003, https://doi.org/10.1007/bf02835237 DOI: https://doi.org/10.1007/BF02835237

Schnatz, K., Nonlinear duality and best approximation in metric linear spaces, J. Approx. Theory, 49, no. 3, pp. 201-218, 1987, https://doi.org/10.1016/0021-9045(87)90099-2 DOI: https://doi.org/10.1016/0021-9045(87)90099-2

Singer, I., Best Approximation in Normed Linear spaces by Elements of Linear subspaces, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New-York-Berlin, 1970, https://doi.org/10.1007/978-3-662-41583-2_4 DOI: https://doi.org/10.1007/978-3-662-41583-2_2

Singer, I., Caracterisations des éléments de la meilleure approximation dans un espace de Banach quelconque, Acta Sci. Math., 17, pp. 181-189, 1956.