Abstract:We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.
MSC:40A35, 41A35, 46E30
ABSTRACT. Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for ( )-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.
ABSTRACT. Some aspects of the theory of order and (D)-convergence in ( )-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.
We prove some equivalence results between limit theorems for sequences of ( )-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same (O)-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.
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