arXiv subjects
Publications and source records attributed to Antonio Boccuto.
The main purpose of this paper is to apply the theory of vector lattices and the related abstract modular convergence to the context of Mellin-type kernels and (non)linear vector lattice-valued operators, following the construction of an integral given in earlier papers.
A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion.
In order to precondition Toeplitz systems, we present a new class of simultaneously diagonalizable real matrices, the Gamma-matrices, which include both symmetric circulant matrices and a subclass of the set of all reverse circulant matrices. We define some algorithms for fast computation of the product between a Gamma-matrix and a real vector and between two Gamma-matrices. Moreover, we illustrate a technique of approximating a real symmetric Toeplitz matrix by a Gamma-matrix, and we show that the eigenvalues of the preconditioned matrix are clustered around zero with the exception of at most a finite number of terms.
We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.
We investigate some main properties of lattice group-valued k-triangular set functions and prove some Brooks-Jewett, Nikodym, Vitali-Hahn-Saks and Schur-type theorems and their equivalence. A Drewnowski-type theorem on existence of continuous restrictions of (s)-bounded set functions is given.
We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.
$L^p$ spaces are investigated for vector lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the vector lattice context, and some properties of the Brownian Motion and the Brownian Bridge are studied, to solve some stochastic differential equations.
Henstock-type integrals are considered, for multifunctions taking values in the family of weakly compact and convex subsets of a Banach lattice $X$. The main tool to handle the multivalued case is a Rådström-type embedding theorem established by C. C. A. Labuschagne, A. L. Pinchuck, C. J. van Alten in 2007. In this way the norm and order integrals reduce to that of a single-valued function taking values in an $M$-space, and new proofs are deduced for some decomposition results recently stated in two recent papers by Di Piazza and Musial based on the existence of integrable selections.
Here we present some comparison results between Birkhoff and McShane multivalued integration.
We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice $X$. Both the single-valued case and the multivalued one are considered (in the last case mainly $cwk(X)$-valued mappings are discussed). The main tool to handle the multivalued case is a Rådström-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an $M$-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selections. Also the order-type integral has been studied: for the single-valued case some basic results from [21] have been recalled, enlightning the differences with the norm-type integral, specially in the case of $L$-space-valued functions; as to multivalued mappings, a previous definition ([6]) is restated in an equivalent way, some selection theorems are obtained, a comparison with the Aumann integral is given, and decompositions of the previous type are deduced also in this setting. Finally, some existence results are also obtained, for functions defined in the real interval $[0,1]$.
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 operatorsin the vector lattice setting, and also for the Brownian motion and related stochastic processes.
A multivalued integral in Riesz spaces is given using the Kurzweil-Henstock integral construction. Some of its properties and a comparison with the Aumann approach are also investigated.