Xenofon Dimitriou

Schur lemma and limit theorems in lattice groups with respect to filters

Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for (ℓ)-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.

Convergence Theorems For Lattice Group-Valued Measures

This chapter contains a historical survey about limit and boundedness theorems for measures since the beginning of the last century. In these kinds of theorems, there are two substantially different methods of proofs: the sliding hump technique and the use of the Baire category theorem. We deal with Vitali-Hahn-Saks, Brooks-Jewett, Nikodým convergence and boundedness theorems, and we consider also some related topics, among which Hahn-Schur-type theorems and some other kind of matrix theorems, the uniform boundedness principle and some (weak) compactness properties of spaces of measures. In this context, the Rosenthal lemma, the biting lemma and the Antosik-Mikusinski-type diagonal lemmas play an important role. We consider the historical evolution of convergence and boundedness theorems for σadditive, finitely additive and non-additive measures, not only real-valued and defined on σ-algebras, but also defined and/or with values in abstract structures.

Abstract Korovkin-type theorems in modular spaces and applications

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.

Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in (l)-groups

ABSTRACT Some Brooks-Jewett, Vitali-Hahn-Saks and Nikod´ym convergence- -type theorems in the context of (l)-groups with respect to ideal convergence are proved. Moreover, an example is given.

Modular filter convergence theorems for abstract sampling type operators

We study the problem of approximating a real-valued function f by considering sequences of general operators of sampling type, which include both discrete and integral ones. This approach gives a unitary treatment of various kinds of classical operators, such as Urysohn integral operators, in particular convolution integrals, and generalized sampling series.

Some versions of limit and Dieudonné-type theorems with respect to filter convergence for (ℓ)-group-valued measures

Some limit and Dieudonné-type theorems in the setting of (ℓ)-groups with respect to filter convergence are proved, extending earlier results.

Ideal Limit Theorems and Their Equivalence in

We prove some equivalence results between limit theorems for sequences of

(\ell)

(\ell)

-Group Setting

-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

Basic matrix theorems for

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