Tauberian theorem

plural Tauberian theorems · alternative tauberian theorem

1. mathematical analysisAny 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.

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

   1957, Einar Hille, Ralph Saul Phillips, Functional Analysis and Semi-groups, Part 1, American Mathematical Society, page 117

   2 more examples

   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.

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