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.
About the Author: Education Universities of Leiden and Utrecht, Mathematics and Physics, 1940--49 (with war-time interruptions)
Ph.D. in Mathematics, Leiden, 1949
Regular professorships (Mathematics)
Technical University Delft (Netherlands), 1951-(Jan)1953
University of Wisconsin (Madison), (Feb)1953--64
(Chairman, Program in Applied Mathematics and Engineering Physics, 1956--61)
University of California San Diego (La Jolla), 1964--74
(Chairman, Dept of Mathematics, 1971--73)
University of Amsterdam, 1974--(Jan)93
(Director, Math. Institute, 1980--83)
Temporary and visiting positions
Mathematical Center, Amsterdam, 1947--49
Purdue University, Acad. yrs 1949--51
University of Michigan, Summer 1950
Stanford University, Acad. yr 1961--62 and several summers
Claremont Graduate School, Sep. 1969 -- Jan. 1970
University of Oregon, Summer 1970
Imperial College, London, Acad. yr 1970--71
Technical University Eindhoven, Summer 1971
California Institute of Technology, Spring 1988
Bar-Ilan University (Israel), Spring 1992
Honors and special assignments
Reynolds' award for outstanding teaching of future engineers, University of Wisconsin, 1956
Elected Fellow Amer. Assoc. Adv. Science, 1961
Chairman, American Mathematical Society Summer Research Institute on "Entire functions and related parts of analysis", La Jolla, 1966
Member, KNAW (Royal Netherlands Academy of Arts and Sciences) since 1975
Honorary doctorate, University of Gothenburg (Sweden), 1978
Chairman, Wiskundig Genootschap (Netherlands Mathematical Society), 1982--84
Lester R. Ford Prize (1987) and Chauvenet Prize (1989) for mathematical exposition (Mathematical Association of America)
Elected honorary member, Netherlands Math. Soc., 1998
Honorary member, Amer. Math. Society
Editor or co-editor of various mathematical journals and of conference proceedings at one time or another
