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Modified Kantorovich-type Sampling Series in Orlicz Space Frameworks

Pooja Gupta

TL;DR

The paper develops a modified Kantorovich-type generalized sampling framework on $\mathbb{R}$ to handle irregular sampling via a kernel-based operator with a nonlinear interval transform $t^\alpha$. It establishes modular convergence in Orlicz spaces $L^{\eta}$ and $L^p$ convergence, and specializes the results to $L^p$, $L\log L$, and exponential Orlicz spaces, including applications to discontinuous functions. The results are complemented by graphical demonstrations comparing the modified operator to classical schemes and by guidance on kernel choices (Fejér, Jackson, and B-spline) to control truncation errors. Overall, the work provides a robust, space- and kernel-aware framework for accurate, irregular sampling on $\mathbb{R}$ with practical implications for signal processing and computational mathematics.

Abstract

This study examines a modified Kantorovich approach applied to generalized sampling series. The paper establishes that the approximation order to a function using these modified operators is atleast as good as that achieved by classical methods by using some graphs. The analysis focuses on these series within the context of Orlicz space \( L^η(\mathbb{R}) \), specifically looking at irregularly spaced samples. This is crucial for real-world applications, especially in fields like signal processing and computational mathematics, where samples are often not uniformly spaced. The paper also establishes a result on modular convergence for functions \( g \in L^η(\mathbb{R}) \), which includes specific cases like convergence in \( L^{p}(\mathbb{R}) \)-spaces, \( L \log L \)-spaces, and exponential spaces. The study then explores practical applications of the modified sampling series, notably for discontinuous functions and provides graphs to illustrate the results.

Modified Kantorovich-type Sampling Series in Orlicz Space Frameworks

TL;DR

The paper develops a modified Kantorovich-type generalized sampling framework on to handle irregular sampling via a kernel-based operator with a nonlinear interval transform . It establishes modular convergence in Orlicz spaces and convergence, and specializes the results to , , and exponential Orlicz spaces, including applications to discontinuous functions. The results are complemented by graphical demonstrations comparing the modified operator to classical schemes and by guidance on kernel choices (Fejér, Jackson, and B-spline) to control truncation errors. Overall, the work provides a robust, space- and kernel-aware framework for accurate, irregular sampling on with practical implications for signal processing and computational mathematics.

Abstract

This study examines a modified Kantorovich approach applied to generalized sampling series. The paper establishes that the approximation order to a function using these modified operators is atleast as good as that achieved by classical methods by using some graphs. The analysis focuses on these series within the context of Orlicz space \( L^η(\mathbb{R}) \), specifically looking at irregularly spaced samples. This is crucial for real-world applications, especially in fields like signal processing and computational mathematics, where samples are often not uniformly spaced. The paper also establishes a result on modular convergence for functions \( g \in L^η(\mathbb{R}) \), which includes specific cases like convergence in \( L^{p}(\mathbb{R}) \)-spaces, -spaces, and exponential spaces. The study then explores practical applications of the modified sampling series, notably for discontinuous functions and provides graphs to illustrate the results.

Paper Structure

This paper contains 6 sections, 11 theorems, 68 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Advanced Generalization of Kantorovich’s Sampling Series
  4. Convergence theorems
  5. Applications to specific classes of Function Spaces
  6. Applications and examples with graphical representations

Key Result

Lemma 3.1

Given the previously stated assumptions about the kernel $\chi$, it can be concluded that

Figures (11)

  • Figure 1: Graph of $F(y) = \frac{1}{2} \text{sinc}^2\left(\frac{y}{2}\right)$
  • Figure 2: Graph of the function $f_2$
  • Figure 3: Comparison of $f_1(x)$, $(S_w^F f_1)(x)$ and $(T_w^F f_1)(x)$ for $w = 5$ and $\alpha = 1/2$
  • Figure 4: Comparison of $f_1(x)$, $(S_w^Ff_1) (x)$ and $(T_w^F f_1)(x)$ for $w = 10$ and $\alpha = 1/2$
  • Figure 5: Comparison of $f_2(x)$, $(S_w^Ff_2) (x)$ and $(T_w^F f_2)(x)$ for $w =5$ and $\alpha = 1/2$
  • ...and 6 more figures

Theorems & Definitions (18)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 8 more