Best uniform approximation of semi-Lipschitz functions by extensions

Costică Mustăţa

doi:10.33993/jnaat362-864

Authors

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

DOI:

https://doi.org/10.33993/jnaat362-864

Keywords:

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

Abstract views: 286

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

E_{d} ({F |}_{Y})

of all extensions of

{F |}_{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, \overset{―}{d})

-sequentially compact, or of finite diameter.

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References

Cobzaş, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27 (3), pp. 275-296, 2004. DOI: https://doi.org/10.2989/16073600409486100

Cobzaş, S., Asymmetric locally convex spaces, Int. J. Math. Math. Sci., 16, pp. 2585-2608, 2005. DOI: https://doi.org/10.1155/IJMMS.2005.2585

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. DOI: https://doi.org/10.2989/16073600309486046

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. DOI: https://doi.org/10.1007/978-94-017-0470-0_3

McShane, E. T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837-842, 1934. DOI: https://doi.org/10.1090/S0002-9904-1934-05978-0

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. DOI: https://doi.org/10.1007/BF01301400

Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000. DOI: https://doi.org/10.1006/jath.1999.3439

Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar., 108(1-2), pp. 55-70, 2005. DOI: https://doi.org/10.1007/s10474-005-0208-9

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. DOI: https://doi.org/10.4995/agt.2005.1956