On the existence and uniqueness of extensions of semi-Holder real-valued functions


Let \((X,d)\) be a quasi-metric space, \(y_{0}\in X\) a fixed element and \(Y\) a subset of \(X\) such that \(y_{0}\in Y\). Denote by \((\Lambda_{\alpha,0}(Y,d),\Vert \cdot|_{Y,d}^{\alpha})\) the asymmetric normed cone of real-valued \(d\)-semi-H\”{o}lder functions defined on \(Y\) of exponent \(\alpha \in(0,1]\), vanishing in \(y_{0}\), and by \((\Lambda_{\alpha,0}(Y,\bar {d}),\Vert \cdot|_{Y,\bar{d}}^{\alpha})\) the similar cone if \(d\) is replaced by conjugate \(\bar{d}\) of \(d\).


Costică Mustăţa
Tiberiu Popoviciu Institute of Numerical Analysis, Romania


Extensions, semi-Lipschitz functions, semi-Holder functions, best approximation, quasi-metric spaces.

Paper coordinates

C. Mustăţa, On the existence and uniqueness of extensions of semi-Holder real-valued functions, Rev. Anal. Numer. Theor. Approx., 39 (2010) no. 2, 134-140.


About this paper


Revue Analysis Numer Theor. Approx.

Publisher Name

Publishing House of the Romanian Academy

Print ISSN


Online ISSN


google scholar link

[1] S. Cobzas, Phelps type duality reuslts in best approximation,Rev. Anal. Numer. Theor.Approx.,31, no. 1., pp. 29–43, 2002.

[2] J. Collins and J. Zimmer, An asymmetric Arzela-Ascoli Theorem, Topology Appl.,154, no. 11, pp. 2312–2322, 2007.

[3] P. Flectherand W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York,1982.

[4] M.G. Kreinand A.A. Nudel’man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow 1973 (in Russian), English translation: American Mathematical Society, Providence, R.I., 1977.

[5] E. Matouskova, Extensions of continuous and Lipschitz functions, Canad. Math. Bull., 43, no. 2, pp. 208–217, 2000.

[6] E.T. McShane, Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.

[7] A. Mennucci, On asymmetric distances, Tehnical report, Scuola Normale Superiore, Pisa, 2004.

[8]C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory,19, no. 3, pp. 222–230, 1977.

[9]C. Mustata,Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal.Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001.

[10]C. Mustata, A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ. Baia Mare, Ser. B. Matematica-Informatica,18, pp. 275–280, 2002.

[11]C. Mustata, Extensions of semi-Holder real valued functions on a quasi-metric space, Rev. Anal. Numer. Theor. Approx., 38, no. 2, pp. 164–169, 2009.

[12] R.R. Phelps,Uniqueness of Hahn-Banach extension and unique best approximation,Trans. Numer. Math. Soc.,95, pp. 238–255, 1960.

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

[14]S. Romaguera and M. Sanchis, Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar,108, nos. 1–2, pp. 55–70, 2005.

[15]J.H. Wells and L.R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975. Received by the editors: April 13, 2010.

Related Posts