General Tauberian Theorems on the Real Line

Abstract

We prove general slow and very slow oscillation Tauberian theorems for the class of all complex, regular summability methods on the real line, under a broad definition which includes the classical methods. This yields short proofs for the best known classical Tauberian theorems, as well as some new interesting concrete theorems, for both positive and non-positive methods. We give a coherent, nearly algorithmic process that explains the origin of the Tauberian conditions, the relation between boundedness and slow and very slow oscillation theorems, or O and o theorems, as well as the hitherto ad hoc manipulations that reduce most of the classical Tauberian theorems to an application of Wiener′s Tauberian theorem. The main underlying idea, of probabilistic inspiration, is the duality between weak star compactness of a family of tightening (affine) scalings of the measures defining the summability method, and the compactness properties for the corresponding family obtained by applying the same scalings to a generalized slowly oscillating Borel measurable function, where the slow oscillation concept is uniquely and naturally determined by the tightness or compactness requirement. The absence of this duality for generalized slowly decreasing Borel measurable functions presents special difficulties, which are, nevertheless, overcome to arrive at general Tauberian theorems for one-sided Tauberian conditions.