Characterization of ε-nearest points in space with asymmetric seminorm

2 years ago

(original), Approximation Theory, asymmetric norm, best approximation, paper

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], [3], [5].

Authors

Costica Mustata
“Tiberiu Popovicu” Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Asymmetric seminormed spaces; ε-nearest points; characterization.

Paper coordinates

C. Mustăţa, Characterization of ε-nearest points in space with asymmetric seminorm, Rev. Anal. Numer. Theor. Approx. 33 (2004) no. 2, 203-208.

PDF

https://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no2-art11/Mustata-pdf

About this paper

Journal

Revue d’Analyse Numer.Theor. Approx.

Publisher Name

Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no2-art11

Print ISSN

2501-059X

Online ISSN

2457-6794

google scholar link

[1] 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.
[2] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx. Theory, 94, no. 1, pp. 54–72, 1998.
[3] Cobzas, S. and Mustata, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Num´er. Th´eor. Approx., 33, no. 1, pp. 39–50, 2004.
[4] Cobzas, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27, pp. 1–22, 2004.
[5] Garcia-Raffi, L. M., Romaguera S. and Sanchez-Perez, E. A., The dual space of an asymmetric normed linear space, Quaest. Math., 26, no. 1, pp. 83–96, 2003.
[6] Garcia-Raffi, L.M., Romaguera S. and Sanchez-Perez, E. A. , On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29, no. 3, pp. 717–728, 2003 (electronic).
[7] 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.
[8] 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.
[9] Mabizela, S., Characterization of best approximation in metric linear spaces, Scientiae Mathematicae Japonica, 57, 2, pp. 233–240, 2003.
[10] Mustata, C. , On the best approximation in metric spaces, Mathematica – Revue d’Analyse Numerique et de Theorie de l’Approximation, L’Analyse Numerique et la Theorie de l’Approximation, 4, pp. 45–50, 1975.
[11] Mustata, C. , On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Numer. Theor. Approx., 32, no. 2,
pp. 187–192, 2003.
[12] Ni, Renxing, Existence of generalized nearest points, Taiwanese J. Math., 7, no. 1, pp. 115–128, 2003.
[13] Pantelidis, G., Approximations theorie f¨ur metrich linear R¨aume, Math. Ann., 184, pp. 30–48, 1969.
[14] Rezapour, Sh., ε-pseudo Chebyshev and ε-quasi Chebyshev subspaces of Banach spaces, Technical Report, Azarbaidjan University of Tarbiot Moallem, 2003.
[15] Rezapour, Sh., ε-weakly Chebyshev subspaces of Banach spaces, Analysis in Theory and Applications, 19, no. 2, pp. 130–135, 2003.
[16] Schnatz, K., Nonlinear duality and best approximation in metric linear spaces, J. Approx. Theory, 49, no. 3, pp. 201–21, 1987.
[17] 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.
[18] Singer, I., Caracterisations des ´el´ements de la meilleure approximation dans un espace de Banach quelconque, Acta Sci. Math., 17, pp. 181–189, 1956.