Weighted Approximation By Max-Product Generalized Exponential Sampling Series (2409.14884v2)

Abstract: In this article, we study the convergence behaviour of the classical generalized Max Product exponential sampling series in the weighted space of log-uniformly continuous and bounded functions. We derive basic convergence results for both the series and study the asymptotic convergence behaviour. Some quantitative approximation results have been obtained utilizing the notion of weighted logarithmic modulus of continuity.

• The paper establishes that Max-Product exponential sampling operators are well-defined and effectively approximate log-uniformly continuous functions in weighted spaces.
• It rigorously proves both pointwise and uniform convergence using weighted modulus of continuity to quantify the rate of approximation.
• The work extends classical exponential sampling theory with non-linear Max-Product techniques, offering new insights for signal processing and optical physics applications.

Weighted Approximation By Maxproduct Generalized Exponential Sampling Series

The paper "Weighted approximation By Maxproduct Generalized Exponential Sampling Series" by Satyaranjan Pradhan investigates the convergence behavior of classical generalized Max Product exponential sampling series within the mathematical framework of weighted spaces, specifically focusing on log-uniformly continuous and bounded functions. This manuscript explores various facets of approximation theory, primarily revolving around Max-Product operators applied to exponentially-spaced sample data, which have significant applications in optical physics, engineering, and signal processing.

Introduction to Exponential and Generalized Exponential Sampling

Sampling and reconstruction of functions is a cornerstone of approximation theory, with foundational contributions from Whittaker, Kotelnikov, and Shannon leading to the WKS sampling theorem for band-limited signals. Ostrowski et al. and later Bertero and Pike extended this framework to exponentially-spaced sampling, resulting in the exponential sampling formula for Mellin band-limited functions. The exponential sampling formula allows for the reconstruction of functions from exponentially-spaced data, crucial for numerous problems in applied sciences.

Pradhan's paper extends the previously established exponential sampling series to a broader class of log-continuous functions using generalized kernels, bypassing the limitation of band-limitedness. This generalization leverages the Mellin analysis, offering a robust framework for handling exponential sampling and approximation problems.

Max-Product Approach and Its Significance

The Max-Product approach, introduced by Coroianu and Gal, deviates from linear counterparts by using maximum or supremum in place of sums or series. This non-linear approach is proposed to yield higher orders of approximation. The current paper applies this technique to exponential sampling operators, thereby generalizing the approximation results to weighted spaces of functions, including a focus on log-uniformly continuous functions.

Key Results and Theorems

The paper is structured around proving several vital properties of Max-Product generalized exponential sampling operators:

1. Well-Definition: The operators are explicitly shown to be well-defined within the log-uniformly continuous and bounded function space. Detailed bounds are derived for the kernel functions and their effects on the operators.
2. Pointwise and Uniform Convergence: It is proven that Max-Product operators converge both point-wise and uniformly for functions in specific weighted spaces. This includes rigorous proofs demonstrating that the Max-Product operators achieve convergence at log-continuity points.
3. Rate of Convergence: Utilizing weighted modulus of continuity, quantitative results on the rates of convergence are derived. These results elucidate how quickly the operators approximate functions within the given function space.
4. Voronovskaja-Type Theorem: A quantitative form of the classical Voronovskaja theorem is established for these operators, providing precise asymptotic behavior of the Max-Product generalized exponential sampling operators.

Implications and Future Directions

The implications of this research are twofold. Practically, the improved convergence properties and higher order of approximation facilitated by the Max-Product approach have significant potential applications in signal processing and related fields where exponentially-spaced data is common. Theoretically, the paper contributes to the broader understanding of non-linear operators in approximation theory, particularly within weighted spaces of functions.

Future research could extend these results to multi-dimensional settings or to other types of non-linear operators. There is also potential for exploring the applications of these theoretical advancements in emerging fields such as neural network approximations, where the Max-Product approach could enhance the performance and accuracy of deep learning models in dealing with corrupted or unevenly-spaced training data.

To summarize, the paper offers valuable insights and significant advancements in the approximation of functions using Max-Product generalized exponential sampling series. It lays a solid groundwork for both practical applications and future theoretical developments in the field of approximation theory.