About the Book
This monograph presents the theory of the Mellin transform and the resulting Mellin analysis in a rigorous and unified manner. Often dismissed as a subordinate topic within Fourier and Laplace transform, it is instead demonstrated here that the theory is completely independent, can be studied within a self-contained framework, and exhibits some typical characteristics.
In addition to highlighting the foundations of the theory, the book addresses applications to certain partial differential equations, sampling theory and numerical quadrature. These applications provide methods which are in turn of interest in various areas of mathematics, science, and engineering.
Each chapter is enriched by numerous references to further literature and potential research directions. Researchers working in this field will gain new insights and appreciate the deserved attention for this underrated topic in harmonic analysis.
Table of Contents:
- 1. Preliminaries.- 2. Polar-Analytic Functions.- 3. The Foundation of Mellin Analysis.- 4. The Mellin Convolution.- 5. The Finite Mellin transform and the Mellin–Fourier Series.- 6. The Mellin Transform in Xc.- 7. The Mellin Transform in Spaces Xp/c for 1 < p ≤ 2.- 8. Mellin Bandlimited Functions and Paley–Wiener Theorems in Mellin Setting.- 9. Mellin Transforms and Fractional Analysis.- 10. The Mellin Distance.- 11. Mellin Transform Methods for Partial Differential Equations.- 12. Exponential Sampling Theory.- 13. Generalized Exponential Sampling Theory.- 14. Applications to Quadrature over the Positive Real Axis.
About the Author :
Carlo Bardaro is professor of mathematics at the DMI of the University of Perugia (Italy). He taught Mathematical Analysis at the University of Rome "La Sapienza" as an associate professor and at the University of Perugia, where, in 2013, he obtained a full professorship. His areas of research comprise operator theory, function spaces, approximation theory, integral transforms, signal analysis. He published 147 research papers, is a co-author of one book on nonlinear integral operators, is co-editor of two recent special issues of Journal of function spaces and Bollettino della Unione Matematica Italiana. He is editor of several mathematical journals. Paul L. Butzer, born in the Rhein-Ruhr district of Germany, had to leave with his parents and brother for Britain in 1937, lived there for four years, traveled on to Montreal, Canada, studying there, then at the University of Toronto, receiving his Masters degree in 1949, his Ph.D 1951. After teaching at McGill University, Montreal, he returned to Germany in 1955, teaching at the universities of Mainz, Freiburg and Wuerzburg. In 1958 he received an offer to join the RWTH AACHEN, as ordinary professor of mathematics ever since 1961, as Chair Professor in Analysis for 32 years, Prof. Emeritus in 1993. His fields of research include real analysis, approximation theory, functional analysis, operator theory, integral transform theory, signal analysis and prediction theory, combinatorial number theory, probability theory, history of mathematics and in certain areas of astronomy. He published ca. 400 research papers, co-authored some 25 books in a variety of fields, including conference proceedings, from Oberwolfach, Aachen, Yugoslavia, Kuwait. According to the ranking of Research.com, he belonged in the year 2023 to the 1460 best scientists worldwide. He was named a 2024 top scholar by Scholar GPS, ranking him within the top 0.5% of all scholars worldwide. For his lifetime career in all fields he was rated #49,965 overall (all fields ), #11,031 in physical science and maths, #1,118 in maths, and #6 in Fourier analysis. Ilaria Mantellini is associate professor of mathematics at the DMI of the University of Perugia (Italy). She obtained the qualification of University Researcher at the University of Perugia in 1997, teaching various courses in mathematical analysis, and subsequently she became associate professor of Mathematical Analysis at the same university in 2022. Her areas of research comprise, operator theory, function spaces, approximation theory, integral transforms, signal analysis. She published 78 research papers and is editor of two mathematical journals. Gerhard Schmeisser studied mathematics and physics at FAU Erlangen (Germany). After passing the state examination in 1964, he received the Dr. rer. nat. degree in 1967 and completed habilitation in 1974. Since 1978 he has been professor of mathematics at FAU and retired in 2004. His various scientific contacts include multiple research stays at Université de Montréal and at Université de Pau et des Pays de l'Adour. His areas of research comprise numerical analysis with emphasis on quadrature and cubature, polynomials, complex analysis, approximation theory, integral transforms and applications of these fields in signal processing. He has published 124 research papers, co-authored two books, a lecture notes and is co-editor of two contributed volumes; he still serves as an editor of a scientific journal.
