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Local/global uniform approximation of real-valued continuous functions

Anthony W. Hager (2011)

Commentationes Mathematicae Universitatis Carolinae

For a Tychonoff space X , C ( X ) is the lattice-ordered group ( l -group) of real-valued continuous functions on X , and C * ( X ) is the sub- l -group of bounded functions. A property that X might have is (AP) whenever G is a divisible sub- l -group of C * ( X ) , containing the constant function 1, and separating points from closed sets in X , then any function in C ( X ) can be approximated uniformly over X by functions which are locally in G . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

Locally solid topological lattice-ordered groups

Liang Hong (2015)

Archivum Mathematicum

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is...

Minimal positive realizations: a survey of recent results and open problems

Luca Benvenuti, Lorenzo Farina (2003)

Kybernetika

In this survey paper some recent results on the minimality problem for positive realizations are discussed. In particular, it is firstly shown, by means of some examples, that the minimal dimension of a positive realization of a given transfer function, may be much “larger” than its McMillan degree. Then, necessary and sufficient conditions for the minimality of a given positive realization in terms of positive factorization of the Hankel matrix are given. Finally, necessary and sufficient conditions...

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