Tauberian Theory: A Century of Developments
Author: Jacob Korevaar
Published by Springer Berlin Heidelberg
ISBN: 978-3-642-05919-3
DOI: 10.1007/978-3-662-10225-1

Table of Contents:

• The Hardy—Littlewood Theorems
• Wiener’s Theory
• Complex Tauberian Theorems
• Karamata’s Heritage: Regular Variation
• Extensions of the Classical Theory
• Borel Summability and General Circle Methods
• Tauberian Remainder Theory

The Hardy-Littlewood Theorems -- Wiener's Theory -- Complex Tauberian Theorems -- Karamata's Heritage: Regular Variation -- Extensions of the Classical Theory -- Borel Summability and General Circle Methods -- Tauberian Remainder Theory -- References -- Index

Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory. There are the spectacular "high-indices" theorems and Karamata's "regular variation", which permeates probability theory. The author presents Gelfand's elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and "circle" methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book

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