Ratio Tauberian Theorems for Positive Functions and Sequences in Banach Lattices

Abstract

We prove two ratio Tauberian theorems and deduce two generalized Tauberian theorems for functions and sequences with values in positive cones of Banach lattices. Two counter-examples are given to show that the hypotheses in the ratio Tauberian theorems are essential. Mathematics Subject Classification (2000). 40E05, 40G05; 47A35; 47D06.

Ratio limit theorems and Tauberian theorems for vector-valued functions and sequences

“…The convolution of a function f and a positive measure (5) for all β > γ > 0 and t > 0; it may happen that s γ t (μ) = ∞ for some t > 0 when 0 < γ < 1.…”

“…[2,Theorem 3.3], [3]). But, as is seen in [5], neither of these two conditions on u is sufficient for a ratio Tauberian theorem to hold, that is, the existence of the Abel ratio limit still does not imply the existence of the Cesàro ratio limit. A ratio Tauberian theorem for positive functions in a Banach lattice and its discrete counterpart have been established in [5] under suitable extra conditions on the scalar function g in the denominator.…”

“…Abel ratio) limit does not guarantee the existence of the Cesàro (Cesàro ratio) limit (see [2, p. 8] (resp. [5]) for examples). The Tauberian theorem of Hardy and Littlewood asserts that if u is bounded, or it is positive in a Banach lattice, then the existence of the Abel limit also implies the existence of the Cesàro limit, and the two limits coincide (cf.…”

“…He was one of the authors in the previous works (see [1,[4][5][6]) and played an essential role in the beginning of this work. To our sorrow he has passed away during this work.…”

Convergence theorems and Tauberian theorems for functions and sequences in Banach spaces and Banach lattices

“…Naturally, we are interested in generalizing the convergence theorems and Tauberian theorems in Sections 2 and 4 to ratio limit theorems and ratio Tauberian theorems for functions in Banach lattices. Results in this respect will appear in [21].…”