Convergence results in Orlicz spaces for sequences of max-product Kantorovich sampling operators

TL;DR

This work extends the convergence theory of max-product Kantorovich sampling operators from classical $L^p$ settings to the broader framework of Orlicz spaces $L^{\varphi}$. A central modular inequality bounds $I^{\varphi}[\lambda(K_n^{\chi}(f)-K_n^{\chi}(g))]$ in terms of $I^{\varphi}$ of $f-g$, enabling a modular convergence theorem: for all $f$ in $L^{\varphi}_+(\Omega)$ there exists $\lambda>0$ with $I^{\varphi}[\lambda(K_n^{\chi}(f)-f)]\to0$ as $n\to\infty$. The theory covers functions on bounded intervals and the real line, and shows that $K_n^{\chi}$ provides robust nonlinear approximation in diverse spaces, including interpolation and exponential spaces, with concrete kernel examples such as Fejér, de la Vallée-Poussin, and B-spline kernels. These results unify and extend known $L^p$ convergence to a wide class of function spaces relevant in signal processing and analysis.

Abstract

In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions defined on both bounded intervals and on the whole real axis has been considered. Here, under suitable assumptions on the kernels, considered in order to define the operators, we are able to establish a modular convergence theorem for these sampling-type operators. As a direct consequence of the main theorem of this paper, we obtain that the involved operators can be successfully used for approximation processes in a wide variety of functional spaces, including the well-known interpolation and exponential spaces. This makes the Kantorovich variant of max-product sampling operators suitable for reconstructing not necessarily continuous functions (signals) belonging to a wide range of functional spaces. Finally, several examples of Orlicz spaces and of kernels for which the above theory can be applied are presented.

Figures (3)

• Figure 1: The Fejér kernel $\mathcal{F}$.
• Figure 2: The de la Vallée-Poussin kernel $\mathcal{P}$.
• Figure 3: The B-spline kernel $M_{3}$.

Theorems & Definitions (24)

• Lemma 2.1: 1, Lemma 2.1
• Lemma 2.2: 1, Lemma 2.2
• Lemma 2.3
• Remark 2.1
• Definition 2.1
• Lemma 2.4: 2, Lemma 2.6
• Theorem 2.1: 2, Theorem 3.1
• Proposition 3.1
• proof
• Theorem 4.1