Gianluca Vinti

In this paper, we investigate the behavior of semi-discrete sampling operators of the Durrmeyer type within the framework of Sobolev–Orlicz spaces. We begin by addressing the problem of modular convergence through a density approach, using the density of compactly supported smooth functions in Sobolev–Orlicz spaces with respect to the modular topology. To this end, we also employ the boundedness of the celebrated Hardy–Littlewood maximal operator in the Orlicz setting. For completeness, we also provide a quantitative analysis to establish sharper estimates based on suitable Orlicz moduli of continuity, in their strong (with respect to the Luxemburg norm) and weak (modular) version. This is obtained through a direct approach based on the Minkowski inequality within the Orlicz framework, involving [ ]-functions with weaker growth condition at infinity. Consequently, Luxemburg norm and modular convergence theorems in broader functional spaces are derived. Throughout the paper, we discuss several examples of functional spaces, including the Sobolev versions of Zygmund and exponential-type spaces.

In this paper we introduce and study the nonlinear version of sampling Durrmeyer operators. For what concerns the space of continuous functions, a pointwise and uniform convergence theorem have been established. Additionally, approximation results for nonlinear sampling Durrmeyer operators in the general setting of Orlicz spaces are also provided. In particular, a general modular convergence theorem has been obtained via a density approach. As a corollary, it follows that this theory applies to particular cases of Orlicz spaces, such as to L ^p − spaces, interpolation spaces ( L ^α log ^β L − spaces) and exponential type spaces. Finally, various applications and examples, including graphical representations, are provided.

In the present paper we study the approximation properties of the Durrmeyer sampling-type operators in the frame of the space of functions of bounded variation. In particular, besides some preliminary results concerning an estimate in variation of the operators in terms of the variation of the function and a regularizing property of the Durrmeyer sampling series, the main result is the convergence in variation by means of the operators. In order to prove it, a relation between the derivative of the Durrmeyer sampling series and a linear combination of the sampling Kantorovich operators applied to a convolution product involving the derivative of the function is exploited. Finally, we obtain a quantitative estimate of the rate of approximation.

In this paper, we introduce and study a class of nonlinear multivariate Durrmeyer sampling operators, extending to a multidimensional framework the univariate nonlinear operators previously investigated in [ 56 ]. Pointwise and uniform convergence results for bounded and, continuous or uniformly continuous functions, respectively, are established, ensuring a solid foundational understanding of the operators' approximation properties. Moving beyond classical settings, we extend our analysis to Orlicz spaces, which provide a suitable framework for handling signals that may not be continuous. Here, we prove a modular convergence theorem using a density argument. Additionally, examples of kernel functions are provided and graphical illustrations that visually demonstrate the convergence and reconstruction behavior of the operators are presented.

This paper presents a study that leverages the important and innovative approximation properties of sampling Kantorovich operators, implemented in the processing of computed tomography (CT) images of patients with abdominal aortic aneurysm (AAA). By exploiting the remarkable reconstruction capabilities of these operators, we investigate a deterministic model grounded in a segmentation algorithm where sampling Kantorovich operators play a crucial role. This mathematically-based approach is compared with an artificial intelligence-based method employing a U-Net neural network. Both methods aim to segment the patent area of the aortic vessel, offering innovative and alternative techniques to the nephrotoxic contrast agents, typically used in diagnosing AAA. The results obtained from testing both methods were evaluated numerically and visually, demonstrating that both approaches yield accurate outcomes.