Weighted approximation By Max-product Kantrovich type Exponential Sampling Series

TL;DR

This work analyzes the convergence and asymptotic behavior of the Max-Product Kantrovich exponential sampling operators in the weighted space of $\log$-uniformly continuous and bounded functions. Using Mellin analysis and a weighted modulus of continuity $\Upsilon(h,\rho)$, the authors establish well-definedness, pointwise and weighted-norm convergence, and a quantitative Voronovskaja-type expansion, with convergence bounds expressed in terms of kernel moments. They provide concrete kernel examples (B-spline, Mellin--Fejér, Mellin--Jackson) and numerical demonstrations illustrating how kernel choice and sampling granularity $m$ influence accuracy. The results extend exponential sampling theory by incorporating max-product nonlinearities to enhance approximation orders in Mellin-analytic settings, with implications for processing exponentially spaced data. Overall, the paper delivers rigorous convergence theory and practical kernel-based guidance for weighted exponential sampling operators.

Abstract

In this study, we examine the convergence characteristics of the Max-Product Kantrovich type exponential sampling series within the weighted space of log-uniformly continuous and bounded functions. The research focuses on deriving fundamental convergence results for the series and analyzing its asymptotic convergence behavior. The study estimates the rate of convergence using the weighted logarithmic modulus of continuity and establishes a quantitative Voronovskaja-type theorem offering insights into the asymptotic behavior of the series. Additionally, we present the example of kernel functions satisfying assumptions of the presented theory along with graphical demonstration and estimates of approximation.

Figures (3)

• Figure 1: Approximation of $h_{1}(z)$ by $\mathscr{M}^{B_{3}}_{m}(h,z)$ for $m=20,50, 100$ .
• Figure 2: Approximation of $h_{2}(z)$ by $\mathscr{M}^{ F_{1}^{0}}_{m}(h,z)$ for $m=20,50,100$.
• Figure 3: Approximation of $h_{3}(z)$ by $\mathscr{M}^{ J^{0}_{1,3}}_{m}(h,z)$ for $m=20,50, 100$.

Theorems & Definitions (11)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Lemma 2.7
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3