Abstract

A set of continuous functions defined on an interval I is called an n-parameter Beckenbach family, if each n points of I×RI \times \mathbb{R} (with pairwise distinct first coordinates) can be interpolated by a unique element of the set. The aim of the present note is to characterize such pairs of real valued functions that can be separated by a member of a given convex Beckenbach family of order n. The key idea of the proof is to identify the family with Rn\mathbb{R}^n via a suitable homeomorphism. Then, the classical Helly Theorem guarantees the existence of a proper separator.

Author information

Contact details are reproduced from the original publication and may be historical.

Mihály Bessenyei

Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary

besse@science.unideb.hu

Keywords

  • Interpolation families
  • Haar and Chebyshev systems
  • Separation theorems
  • Helly's Theorem

Mathematics Subject Classification

26A51; 39B62, 52A20

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Suggested citation

M. Bessenyei, P. Szokol. “Separation by Convex Interpolation Families.” Journal of Convex Analysis 20 (2013), No. 4, 937–946. https://doi.org/10.68381/jca20050

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.