On the Modification of Mellin Convolution Operator and Its Associated Information Potential

Abstract

In this paper, we define a new generalization of Mellin-Gauss-Weierstrass operators that preserve logarithmic functions. We compute logarithmic moments of the new operators and describe the behavior of the modified operators in some weighted spaces. The weighted approximation properties of operators including weighted approximation and weighted quantitative type approximation properties, using weighted logarithmic modulus of continuity, are presented. Using the Mellin-Gauss-Weierstrass kernel $p(·,·)$ as a logarithmic probability density, we study the associated information potential, the expected value $E[\log \hspace{0.17em}p(·,·)]$ and the variance $\mathit{Var}[\log \hspace{0.17em}p(·,·)]$.

KEYWORDS:

• Information potential
• logarithmic moments
• Mellin-Gauss-Weierstrass operators
• weighted logarithmic modulus of continuity

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 41A36
• Secondary: 41A25

Acknowledgments

The authors are very grateful to the reviewers for their comments, remarks and suggestions which greatly improved an initial version of the paper.

Disclosure statement

The authors declare that they have no conflict of interest.

Data availability statement

This manuscript has no associated data.

Additional information

Funding

The work of the second author was supported by Lucian Blaga University of Sibiu (Knowledge Transfer Center) & Hasso Plattner Foundation research grants LBUS-HPI-ERG-2023-01.