Max-product Kantorovich sampling operators: quantitative estimates in functional spaces
Lorenzo Boccali, Danilo Costarelli, Gianluca Vinti
TL;DR
The paper addresses the rate of modular approximation by max-product Kantorovich sampling operators $K_{n}^{\chi}$ in Orlicz spaces $L^{\varphi}(\Omega)$. The authors develop Jackson-type estimates using the Orlicz-type modulus of smoothness $\omega(f,\delta)_{\varphi}$ and a kernel-decay condition $(\chi4)$ to obtain explicit modular error bounds. For compact domains, a Peetre-type Orlicz K-functional bound $\mathcal{K}(f,\lambda,\delta)_{\varphi}$ yields a modular error bound of the form $I^{\varphi}[\lambda_{1}(K_{n}^{\chi}(f)-f)] \le A_{1}\mathcal{K}(f,\lambda_{0},A_{2}/n)_{\varphi}$ with explicit constants. The results cover a wide class of functional spaces (e.g., $L^{p}$, interpolation, exponential spaces) and imply Lipschitz-class convergence rates, providing a unified framework for non-linear max-product sampling in approximation theory.
Abstract
In this paper, we study the order of approximation for max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as $L^{p}$-spaces, interpolation spaces and exponential spaces.