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.

Abstract: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. MSC:40A35, 41A35, 46E30

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

Abstract. Here we give a new approach to the Paley-Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar-analytic function in the Mellin setting and Mellin-Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.

Abstract. In this paper we establish a version of the Paley-Wiener theorem of Fourier analysis in the frame of the Mellin transform. 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.