Locally piecewise affine functions and their order structure

2016/03/15 by Samer Adeeb, Adeeb, Samer, Vladimir G. Troitsky +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #math.FA #msc:46A40 #msc:46E05

paper · pdf · doi:10.48550/arxiv.1603.04897

11 pages

arxiv created 2016/03/15 · arxiv updated 2016/03/17

Abstract

Piecewise affine functions on subsets of \mathbb R^m were studied in \citeOvchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07. In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in C(\mathbb R^m), while piecewise affine functions are sequentially order dense in C(\mathbb R^m). This paper is partially based on \citeAdeeb:14.

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