Best uniform approximation of semi-Lipschitz functions by extensions

Costică Mustăţa

Authors

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

Keywords:

semi-Lipschitz functions, uniform approximation, extensions of semi-Lipschitz functions

Abstract views: 219

Abstract

In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function \(F\) defined on an asymmetric metric space \((X,d),\) by the elements of the set \(\mathcal{E}_{d}(\left. F\right\vert _{Y})\) of all extensions of \(\left.F\right\vert _{Y}\) \((Y\subset X),\) preserving the smallest semi-Lipschitz constant. It is proved that this problem has always at least a solution, if \((X,d)\) is \((d,\overline{d})\)-sequentially compact, or of finite diameter.

Downloads

Download data is not yet available.

References

Cobzaş, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27 (3), pp. 275-296, 2004.

Cobzaş, S., Asymmetric locally convex spaces, Int. J. Math. Math. Sci., 16, pp. 2585-2608, 2005.

Cobzaş, S. and Mustăţa, C., Best approximation in spaces with asymmetric norm,. Rev. Anal. Numér. Thèor. Approx., 33 (1), pp. 17-31, 2006, http://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no1-art4

Cobzaş, S. and Mustăţa, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numér. Théor. Approx., 31, 1, pp. 35-50, 2004, http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art5

Collins, J. and Zimmer, J., An asymmetric Arzelà-Ascoli theorem, http://bath.ac.uk/math-sci/BICS, Preprint, 16, 12 pp, 2005.

Garcia-Raffi, L. M., Romaguera, S. and Sánchez-Pérez, E. A., The dual space of an asymmetric linear space, Quaest. Math., 26, pp. 83-96, 2003.

Künzi, H. P. A., Nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topologies, in: Handbook of the History of General Topology, ed. by C.E. Aull and R. Lower, 3, Hist. Topol. 3, Kluwer Acad. Publ. Dordrecht, pp. 853-968, 2001.

McShane, E. T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837-842, 1934.

Menucci, A., On asymmetric distances, Technical Report, Scuola Normale Superiore, Pisa, 2004.

Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Theor. Approx, 30, 1, pp. 61-67, 2001, http://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art8

Mustăţa, C., On the extremal semi-Lipschitz functions, Rev. Anal. Numér. Theor. Approx, 31, 1, pp. 103-108, 2002, http://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art11

Mustăţa, C., On the approximation of the global extremum of a semi-Lipschitz function, IJMMS (to appear).

Reilly, I.L., Subrahmanyam, P. V. and Vamanamurthy, M. K., Cauchy sequences in quasi-pseudo-metric spaces, Mh. Math., 93 , pp. 127-140, 1982.

Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000.

Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar., 108(1-2), pp. 55-70, 2005.

Romaguera, S., Sánchez-Álvarez, J.M. and Sanchis, M., El espacio de funciones semi-Lipschitz, VI Jornadas de Matematica Aplicada, Universiadad Politécnica de Valencia, pp. 1-15, 2005.

Sánchez-Álvarez, J. M., On semi-Lipschitz functions with values in a quasi-normed linear space, Applied General Topology, 6, 2, pp. 216-228, 2005.