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On the Convergence of Max-product and Max-Min Durrmeyer-type Exponential Sampling Operators

Satyaranjan Pradhan, Abhishek Senapati, Madan Mohan Soren

TL;DR

The paper addresses convergence and approximation properties of Durrmeyer-type Max-product and Max-min exponential sampling operators in the Mellin setting. It defines nonlinear max-based operators $\mathscr{D}^{M}_{n,\Phi,\Psi}$ and $\mathscr{D}^{m}_{n,\Phi,\Psi}$ using kernel pairs $(\Phi,\Psi)$, and proves pointwise and uniform convergence on the log-uniformly continuous bounded function space $\mathscr{LU}_{b}(\mathbb{R}_{+})$, with rates via the logarithmic modulus of continuity. The authors derive quantitative error bounds, including an $O(n^{-\alpha/(1+\alpha)})$ rate for functions in $\mathcal{L}_{\log}^{\alpha}$, and demonstrate the theory with Mellin B-spline and Mellin-Fejér kernels through numerical experiments. Results indicate the Max-product variant offers slightly faster convergence than the Max-min variant, with potential applications in signal and image processing involving exponentially spaced sampling.

Abstract

This article discusses the convergence properties of the Max Product and Max Min variants of Durrmeyer type exponential sampling series. We first establish pointwise and uniform convergence of both operators in the space of log uniformly continuous and bounded functions. The rates of convergence are then analyzed in terms of the logarithmic modulus of continuity. Additionally, the approximation errors of the proposed operators are examined using a variety of kernel functions. Finally, graphical illustrations are provided to demonstrate the convergence behavior of both operators.

On the Convergence of Max-product and Max-Min Durrmeyer-type Exponential Sampling Operators

TL;DR

The paper addresses convergence and approximation properties of Durrmeyer-type Max-product and Max-min exponential sampling operators in the Mellin setting. It defines nonlinear max-based operators and using kernel pairs , and proves pointwise and uniform convergence on the log-uniformly continuous bounded function space , with rates via the logarithmic modulus of continuity. The authors derive quantitative error bounds, including an rate for functions in , and demonstrate the theory with Mellin B-spline and Mellin-Fejér kernels through numerical experiments. Results indicate the Max-product variant offers slightly faster convergence than the Max-min variant, with potential applications in signal and image processing involving exponentially spaced sampling.

Abstract

This article discusses the convergence properties of the Max Product and Max Min variants of Durrmeyer type exponential sampling series. We first establish pointwise and uniform convergence of both operators in the space of log uniformly continuous and bounded functions. The rates of convergence are then analyzed in terms of the logarithmic modulus of continuity. Additionally, the approximation errors of the proposed operators are examined using a variety of kernel functions. Finally, graphical illustrations are provided to demonstrate the convergence behavior of both operators.
Paper Structure (6 sections, 16 theorems, 73 equations, 6 figures, 4 tables)

This paper contains 6 sections, 16 theorems, 73 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Max-product Durrmeyer type Exponential sampling operators
  4. The Max-Min Durrmeyer type Exponential sampling operators
  5. Example Kernels and Numerical Illustration
  6. Concluding Remarks:

Key Result

Lemma 1

(seeAng1) Let $\Phi$ be a bounded function satisfying the condition $(\Phi.{1})$, and let $\mu > 0$. Then

Figures (6)

  • Figure 1: Approximation of $f$ using the Max-Product Durrmeyer operator for different values of $n$.
  • Figure 2: Approximation of $f$ using the Max-Min Durrmeyer operator for different values of $n$.
  • Figure 3: Comparison of the approximations of $f$ by Max-Product and Max-Min Durrmeyer operator.
  • Figure 4: Approximation of $g$ using the Max-Product Durrmeyer operator for different values $n$.
  • Figure 5: Approximation of $g$ using the Max-Min Durrmeyer operator for different values of $n$.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 22 more