203 - Österreichische Mathematische Gesellschaft

J. Korevaar: Tauberian Theory. A Century of Developments. (Grundlehren der
mathematischen Wissenschaften 329.) Springer, Berlin, Heidelberg, New York,
2004, XV+483 S. ISBN 3-540-21058-X H/b 89,95.
What has started as an interesting observation on a converse to some statements
about summability, made by the Austrian mathematician Tauber in 1897, turned
into a veritable and important theory. Typically a Tauberian Theorem states that
the asymptotic behavior of certain “moving averages” is comparable for a large
family of convolution kernels. In combination with certain “slow variation conditions”
this allows to conclude from the existence of limits of those averaged
functions even that the function itself must have the same limit.
One cannot do justice to a big volume like this by listing all the important results
provided. It is certainly the reference work in the fields for the years to come,
hopefully inspiring continued research in this field, which is today still an active
research area (e.g. in the context of locally compact groups). Let us just list the
chapter headers, indicating the important mathematicians who have contributed to
the field: (1) The Hardy-Littlewood Theorems; (2) Wiener’s Theory; (3) Complex
Tauberian Theorems; (4) Karamata’s Heritage: Regular Variation; (5) Extensions
of the Classical Theory; (6) Borel Summability and general Circle Methods; (7)
Tauberian Remainder Theory, as well as almost 50 pages of references. The author
has really done an excellent job in compiling classical and modern theory.
This reviewer is only sorry that his own contribution from 1987 on “An elementary
approach to Wiener’s third Tauberian theorem for the Euclidean n-space” did
not make it into this excellent book [In: Harmonic analysis, symmetric spaces
and probability theory, Cortona/Italy 1984, Symp. Math. 29, 267–301 (1987)]. It
shows the validity of Wiener’s Tauberian theorem for functions of bounded p-
means (not just p = 2) of several variables for p ∈ [1,∞], using Banach algebra
techniques, very much in the spirit of N. Wiener.
H. G. Feichtinger (Wien)
E. Kunz: Introduction to Plane Algebraic Curves. Translated from the original
German by R. G. Belshoff. Birkhäuser, Boston, Basel, Berlin, 2005, XII+293 S.
ISBN 0-8176-4381-8 P/b 52,80.
The present book is a translation of the German lecture Notes (University of Regensburg)
”
Ebene algebraische Kurven“ of the author which appeared 1991. The
main goal is to present commutative ring theory together with algebraic curves as
one of the important applications. The presentation of plane algebraic curves in
this book stresses the algebraic point of view. Concerning topological and analytical
aspects the reader is refered to the rich literature (e.g. Brieskorn-Knörrer:
Plane algebraic curves, Birkhäuser 1986). The book is organized in a way that
the algebraic facts are collected in eleven appendices which cover about one third
of the book. The chapters are: Affine Algebraic Curves, Projective Algebraic
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