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.

Autorentext

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

Inhalt

I The Hardy-Littlewood Theorems.- II Wiener's Theory.- III Complex Tauberian Theorems.- IV Karamata's Heritage: Regular Variation.- V Extensions of the Classical Theory.- VI Borel Summability and General Circle Methods.- VII Tauberian Remainder Theory.- References.