Tauberian-Type Results for Convolution of Sequences

Abstract

Copson (1970) proved a theorem expressible as: If p is a finite strictly decreasing positive sequence, if a is a real semi-infinite bounded sequence, and if a*p is monotone, then a is convergent (the monotonicity of a*p is equivalent to a linear inequality on the a_n). Later Borwein and Russell independently published necessary and sufficient conditions for the conclusion, related to the distribution of the zeros of the generating power series (or polynomial) p(z). Here we use bi-infinite sequences and a sequence space λ with the property (u∈λ, v∈ℓ¹) ⇒ u*v ∈ λ, and we ask for a theorem of the form a*p ∈ λ ⇒ a ∈ λ, provided that the tauberian condition a ∈ ℓ^∞ holds. While (for λ = bv) this already includes all previous results, the theorem can be further improved for semi-infinite sequences.