Tauberian theorem

English

Alternative forms

• tauberian theorem

Etymology

After Austrian and Slovak mathematician Alfred Tauber (1866-1942).

Noun

Tauberian theorem (plural Tauberian theorems)

1. (mathematical analysis) Any of a class of theorems which, for a given Abelian theorem, specifies conditions such that any series whose Abel sums converge (as stipulated by the Abelian theorem) is in fact convergent.
   • 1957, Einar Hille, Ralph Saul Phillips, Functional Analysis and Semi-groups, Part 1, American Mathematical Society, page 117:

     Paragraph five deals with (A*)-algebras and contains a proof of a vector-valued variant of Wiener's Tauberian theorem.

   • 1988, Staff writer, Foreword, [1933, Norbert Wiener, The Fourier Integral and Certain of Its Applications], Cambridge University Press, 1988 reissue, page xi,

     Not only did the general Tauberian theorem give a unifying view on questions involving summations and limits, but it introduced a paradigm for what was called abstract harmonic analysis a few years later. […] Generalized harmonic analysis is the subject-matter of the last chapter, though it was conceived before the Tauberian theorems.

   • 2000, Johann Boos, F. Peter Cass, Classical and Modern Methods in Summability, Oxford University Press, page 167:

     We should emphasize that our main concern is — besides the presentation of Tauberian theorems in the case of special summability methods — to put, by way of examples, different methods in the hands of the reader to prove Tauberian theorems in the case of special summability methods.

Usage notes

G. H. Hardy describes Tauberian theorems as corrected forms of the false converse of Abelian theorems.

Coordinate terms

• Abelian theorem

Further reading

• Divergent series on Wikipedia.Wikipedia
• Hardy–Littlewood tauberian theorem on Wikipedia.Wikipedia
• Tauberian Theorem, Eric W. Weisstein, MathWorld - A Wolfram Web Resource