Antonio Boccuto

Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes☆

A Vitali-type theorem for vector lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operators in the vector lattice setting, and also for the Brownian motion and related stochastic processes.

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.

Some inequalities in classical Spaces with mixed norms

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from Lp. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that Lp is the only space where it is possible to change this order.

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.

Integral and ideals in Riesz spaces

A convergence in Riesz spaces is given axiomatically. A Bochner-type integral for Riesz space-valued functions is introduced and some Vitali and Lebesgue dominated convergence theorems are proved. Some properties and examples are investigated.

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.

A NOTE ON COMPARISON BETWEEN BIRKHOFF AND MCSHANE-TYPE INTEGRALS FOR MULTIFUNCTIONS

Here we present some comparison results between Birkho and McShane multivalued integration.

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.