Meaning of Tauberian theorem | Babel Free

Definitions

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.

Examples

“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.”

“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.”

CEFR level