Tauberian Theory - A Century of Progress

Author: Kreyszig
Publisher:
Publish Date: 2007-01-01
Features: Tauberian Theory / Comparison of different methods for determining the summability of series and integrals, identifying when they converge, providing asymptotic estimates and remainder estimates. Starting from the original origins of Tauberian theory, the author introduces the development of the theory: his professional reviews recreate the excitement brought by early results; discusses the difficult yet fascinating Hardy-Littlewood theorem and its unexpected elegant proof; highly praises Wiener's breakthrough based on Fourier theory, the intriguing high-index "rdquo; theorem, and the Karamata regular variation theory applied to probability theory. The author also mentions Gelfand's algebraic treatment of Wiener's theory and his own distribution method. The book introduces a unified new theory of the Borel method and the "circle" method, and also discusses various Tauberian methods for studying the Prime Number Theorem. At the end of the book, there are numerous references and a detailed index. The book provides a comprehensive discussion of convergence, remainder estimates, and other aspects. Following the author's unique insights into this knowledge, it explains the properties of those very difficult theorems in an accessible and easy-to-understand manner, making the presentation truly beautiful.