Abstract
A theorem of Baire concerning the approximation of lower semicontinuous real valued functions defined on a metric space, by increasing sequences of continuous functions is extended to the “nonsymmetric” case, i.e. for quasi-metric spaces.
Authors
Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis , Cluj-Napoca, Romania
Keywords
Quasi-metric space, semi-Lipschitz function, approximation.
Paper coordinates
C. Mustăţa, On a theorem of Baire about lower semi-continuous functions, Rev. Anal. Numer. Theor. Approx. 37 (2008) no. 1, 71-75.
About this paper
Journal
Revue d’Analyse Numer Theorie Approximation
Publisher Name
Publisher House of the Romanian Academy
Print ISSN
Online ISSN
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[1] Baire, R.,Lecon sur les fonctions discontinues,Paris, Collection Borel, 1905, pp. 121–123.
[2] Borodin, P.A.,The Banach-Mazur theorem for spaces with asymmetric norm and itsapplications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[3] Collins, J.and Zimmer, J.An asymmetric Arzela-Ascoli theorem,Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
[4] K unzi, H.P.A., Nonsymmetric distances and their associated topologies: About theorigin of basic ideas in the area of asymmetric topology, in: Handbook of the Historyof General Topology, edited. by C.E. Aull and R. Lower, vol.3, Kluwer Acad. Publ.,Dordrecht, pp. 853–968, 2001.
[5] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[6] Menucci, A., On asymmetric distances,Technical Report, Scuola Normale Superiore,Pisa, 2004.
[7] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnica, Bucharest, p. 119,1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., EdituraUniversitatii A.I. Cuza Iasi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions. Acta Math. Hungar.,108, no. 1-2, pp. 55–70, 2005.
[2] Borodin, P.A.,The Banach-Mazur theorem for spaces with asymmetric norm and itsapplications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[3] Collins, J.and Zimmer, J.An asymmetric Arzela-Ascoli theorem,Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
[4] K unzi, H.P.A., Nonsymmetric distances and their associated topologies: About theorigin of basic ideas in the area of asymmetric topology, in: Handbook of the Historyof General Topology, edited. by C.E. Aull and R. Lower, vol.3, Kluwer Acad. Publ.,Dordrecht, pp. 853–968, 2001.
[5] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[6] Menucci, A., On asymmetric distances,Technical Report, Scuola Normale Superiore,Pisa, 2004.
[7] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnica, Bucharest, p. 119,1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., EdituraUniversitatii A.I. Cuza Iasi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions. Acta Math. Hungar.,108, no. 1-2, pp. 55–70, 2005.