Carlo Bardaro

Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities

On etablit des inegalites minimax generalisees pour des fonctions prenant des valeurs dans des espaces vectoriels ordonnes

Nonlinear integral operators and applications

In 1903 Fredholm published his famous paper on integral equations. Since then linear integral operators have become an important tool in many areas, including the theory of Fourier series and Fourier integrals, approximation theory and summability theory, and the theory of integral and differential equations. As regards the latter, applications were soon extended beyond linear operators. In approximation theory, however, applications were limited to linear operators mainly by the fact that the notion of singularity of an integral operator was closely connected with its linearity. This book represents the first attempt at a comprehensive treatment of approximation theory by means of nonlinear integral operators in function spaces. In particular, the fundamental notions of approximate identity for kernels of nonlinear operators and a general concept of modulus of continuity are developed in order to obtain consistent approximation results. Applications to nonlinear summability, nonlinear integral equations and nonlinear sampling theory are given. In particular, the study of nonlinear sampling operators is important since the results permit the reconstruction of several classes of signals. In a wider context, the material of this book represents a starting point for new areas of research in nonlinear analysis. For this reason the text is written in a style accessible not only to researchers but to advanced students as well.

Prediction by Samples From the Past With Error Estimates Covering Discontinuous Signals

There are several reasons why the classical sampling theorem is rather impractical for real life signal processing. First, the sinc-kernel is not very suitable for fast and efficient computation; it decays much too slowly. Second, in practice only a finite number N of sampled values are available, so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for N¿ ¿, due to the first drawback. Third, band-limitation is a definite restriction, due to the nonconformity of band and time-limited signals. Further, the samples needed extend from the entire past to the full future, relative to some time t = t0. This paper presents an approach to overcome these difficulties. The sinc-function is replaced by certain simple linear combinations of shifted B-splines, only a finite number of samples from the past need be available. This deterministic approach can be used to process arbitrary, not necessarily bandlimited nor differentiable signals, and even not necessarily continuous signals. Best possible error estimates in terms of an Lp-average modulus of smoothness are presented. Several typical examples exhibiting the various problems involved are worked out in detail.

Applications of the generalized Knaster-Kuratowski-Mazurkiewicz Theorem to variational inequalities

On presente des versions generalisees des resultats de M. Lassonde sur des inegalites de type variationnel en utilisant certaines extensions du theoreme de Knaster-Kuratowski-Mazurkiewicz

Approximation properties in abstract modular spaces for a class of general sampling-type operators

In this article we study approximation properties for the class of general integral operators of the form where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of closed subsets of G with suitable properties and, for every w>0, μ Hw is a regular measure on Hw We give pointwise, uniform and modular convergence theorems in abstract modular spaces and we apply the results to some kind of discrete operators including the sampling-type series.

Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems

Here we give some approximation theorems concerning pointwise convergence for nets of nonlinear integral operators of the form: where the kernel (K λ)λ∈Λ satisfies some general homogeneity assumptions. Here Λ is a nonempty set of indices provided with a topology.

On pointwise convergence of linear integral operators with homogeneous kernels

Here we give some approximation theorems concerning pointwise convergence and rate of pointwise convergence for non-convolution type linear operators of the form: with kernels satisfying some general homogeneity assumptions. Here Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology.

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.

A Korovkin theorem in multivariate modular function spaces

In this paper a modular version of the classical Korovkin theorem in multivariate modular function spaces is obtained and applications to some multivariate discrete and integral operators, acting in Orlicz spaces, are given.

On Convergence Properties for a Class of Kantorovich Discrete Operators

Here we give some pointwise convergence theorems and asymptotic formulae of Voronovskaja type for a general class of Kantorovich discrete operators. Applications to the Kantorovich version of some discrete operators are given.