Best approximation in spaces with asymmetric norm


In this paper we shall present some results on spaces with asymmetric seminorms, with emphasis on best approximation problems in such spaces.


Stefan Cobzas
Babes-Bolyai University, Cluj-Napoca, Romania

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


Spaces with asymmetric norm; best approximation; Hahn-Banach theorem; characterization of best approximation.

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S. Cobzas, C. Mustăţa, Best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx. 35 (2006) no. 1, 17-31.


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Revue d’Analyse Numer. Theor. Approx.

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[1] Alegre, C.Ferrer, J. andGregori, V.,Quasi-uniformities on real vector spaces,Indian J. Pure Appl. Math., 28, no. 7, pp. 929–937, 1997.
[2]___,On the Hahn-Banach theorem in certain linear quasi-uniform structures, ActaMath. Hungar.,82, no. 4, pp. 325–330, 1999.
[3] Alimov, A. R.,The Banach-Mazur theorem for spaces with nonsymmetric distance,Uspekhi Mat. Nauk, 58, no. 2, pp. 159–160, 2003.
[4] Babenko, V. F.,Nonsymmetric approximations in spaces of summable functions,Ukrain. Mat. Zh., 34, no. 4, pp. 409–416, 538, 1982.
[5]___,Nonsymmetric extremal problems of approximation theory, Dokl. Akad. NaukSSSR,269, no. 3, pp. 521–524, 1983.
[6]___,Duality theorems for certain problems of the theory of approximation, Currentproblems in real and complex analysis, Akad. Nauk Ukrain. SSR Inst. Mat., Kiev,pp. 3–13, 148, 1984.
[7] Borodin, P. A.,The Banach-Mazur theorem for spaces with an asymmetric norm andits applications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[8] Cobzas, S.,Phelps type duality results in best approximation, Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.
[9]___,Separation of convex sets and best approximation in spaces with asymmetricnorm, Quaest. Math.,27, no. 3, 275–296, 2004.
[10]___,Asymmetric locally convex spaces,Int. J. Math. Math. Sci., no. 16, 2585–2608,2005.
[11] Cobzas, S. and Mustata, C.,Extension of bilinear functionals and best approximationin 2-normed spaces, Studia Univ. Babes-Bolyai, Mathematica, 43, pp. 1–13, 1998.
[12]___,Extension of bounded linear functionals and best approximation in spaces withasymmetric norm, Rev. Anal. Numer. Theor. Approx.,33, no. 1, pp. 39–50, 2004.
[13] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx.Theory,94, no. 1, pp. 54–72, 1998.
[14] Dolzhenko, E. P. andSevastyanov, E. A.,Approximations with a sign-sensitiveweight (existence and uniqueness theorems), Izv. Ross. Akad. Nauk Ser. Mat.,62, no. 6,pp. 59–102,1998.
[15]___,Sign-sensitive approximations, J. Math. Sci. (New York),91, no. 5, pp. 3205–3257, 1998.
[16]___,Approximation with a sign-sensitive weight (stability, applications to snake the-ory and Hausdorff approximations), Izv. Ross. Akad. Nauk Ser. Mat.,63, no. 3, pp.77–118, 1999.
[17] Ferrer, J.,Gregori, V. andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, no. 3, pp. 877–884, 1993.
[18] Garcıa-Raffi, L. M.,Romaguera, S. and Sanchez Perez, E. A.,Extensions of asymmetric norms to linear spaces, Rend. Istit. Mat. Univ. Trieste,33, nos. 1–2, 113–125, 2001.
[19]___,The bicompletion of an asymmetric normed linear space, Acta Math. Hungar.,97, no. 3, pp. 183–191, 2002.
[20]___,Sequence spaces and asymmetric norms in the theory of computational com-plexity, Math. Comput. Modelling,36, nos. 1–2, pp. 1–11, 2002.
[21]___,The dual space of an asymmetric normed linear space, Quaest. Math.,26, no. 1,pp. 83–96, 2003.
[22]___,On Hausdorff asymmetric normed linear spaces, Houston J. Math.,29, no. 3,pp. 717–728 (electronic), 2003.
[23] Krein, M. G. and Nudelman, A. A., The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian), English translation: American Mathe-matical Society, Providence, R.I., 1977.
[24] Chong Li, On wel l posed generalized best approximation problems, J. Approx. Theory,107, no. 1, pp. 96–108, 2000.
[25] Chong Liand Renxing Ni, Derivatives of generalized distance functions and existenceof generalized nearest points, J. Approx. Theory,115, no. 1, pp. 44–55, 2002.
[26] Mohebi, H.,On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory,107,no. 1, pp. 87–95, 2000.
[27]___,Pseudo-Chebyshev subspaces inL1, Korean J. Comput. Appl. Math.,7, no. 2,pp. 465–475, 2000.
[28]___,On pseudo-Chebyshev subspaces in normed linear spaces, Math. Sci. Res. Hot-Line,5, no. 9, pp. 29–45, 2001.
[29]___,Quasi-Chebyshev subspaces in dual spaces, J. Nat. Geom.,20, nos. 1-2, pp.33–44, 2001.
[30]___,On pseudo-Chebyshev subspaces in normed linear spaces, J. Nat. Geom.,24,nos. 1–2, pp. 37–56, 2003.
[31] Mohebi, H. andRezapour,Sh.,On weak compactness of the set of extensions of acontinuous linear functional, J. Nat. Geom.,22, nos. 1–2, pp. 91–102, 2002.
[32] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[33]___,On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx.,31, no. 1, pp. 103–108, 2002.
[34]___,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ.Baia Mare, Ser. B, Matematica-Informatica,18, no. 2, pp. 275–280, 2002.
[35]___,On the uniqueness of the extension and unique best approximation in the dual ofan asymmetric linear space, Rev. Anal. Numer. Theor. Approx.,32, no. 2, pp. 187–192,2003.

[36] Renxing Ni,Existence of generalized nearest points, Taiwanese J. Math.,7, no. 1, pp.115–128, 2003.
[37] Phelps, R. R.,Uniqueness of Hahn-Banach extensions and best approximations, Trans.Amer. Marth Soc.,95, pp. 238–255, 1960.
[38] Rezapour, Sh.,∈-weakly Chebyshev subspaces of Banach spaces, Anal. Theory Appl.,19, no. 2, pp. 130–135, 2003.
[39] Romaguera, S. and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, no. 2, pp. 292–301, 2000.
[40] Romaguera, S. and Schellekens, M.,Duality and quasi-normability for complexityspaces, Appl. Gen. Topol.,3, no. 1, pp. 91–112, 2002.
[41] Simonov, B. V.,On the element of best approximation in spaces with nonsymmetricquasinorm, Mat. Zametki,74, no. 6, pp. 902–912, 2003.
[42] Singer, I.,Best Approximation in Normed Linear Spaces by Elements of Linear Sub-spaces, Editura Academiei Romane and Springer-Verlag, Bucharest-New York-Berlin,1970.
[43] Zanco, C. and Zucchi, A.,Moduli of rotundity and smoothness for convex bodies,Bolletino U. M. I., (7),7-B, pp. 833–855, 1993

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