Ilaria Mantellini

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.

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.

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.

On Mellin convolution operators: a direct approach to the asymptotic formulae

Here, using Mellin derivatives, a different notion of moment and a suitable modulus of continuity, we state a quantitative Voronovskaja approximation formula for a general class of Mellin convolution operators. This gives a direct approach to the study of pointwise approximation of such operators, without using the Fourier analysis and its results. Various applications to important specific examples are given.

Generalized Sampling Approximation of Bivariate Signals: Rate of Pointwise Convergence

We study a class of bivariate generalized sampling operators and we give a general asymptotic formula for the pointwise convergence. Moreover we study a quantitative version.

A note on the Voronovskaja theorem for Mellin–Fejer convolution operators

Abstract Here, using Mellin derivatives and a different notion of moment, we state a Voronovskaja approximation formula for a class of Mellin–Fejer type convolution operators. This new approach gives direct and simple applications to various important specific examples.

On the Paley-Wiener theorem in the Mellin transform setting

In this paper we establish a version of the Paley-Wiener theorem of Fourier analysis in the frame of Mellin transforms. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.

On Voronovskaja formula for linear combinations of Mellin–Gauss–Weierstrass operators

Abstract Here we give a Voronovskaja formula of high order for linear combinations of the convolution operators ( G w , r f ) ( s ) = ∫ 0 + ∞ ∑ j = 1 r α j K jw ( t ) f ( st ) dt t , where K w is the Mellin–Gauss–Weierstrass kernel. This kind of operator provides a better order of pointwise approximation and leads to asymptotic formulae of type lim w → + ∞ w ν [ ( G w , r f ) ( s ) - f ( s ) ] = A ( f , ν ) where r , ν ∈ N and A ( f , ν ) is a differential operator containing the derivatives of f up to order ν .