Ideal extensions of Olivier's theorem

Abstract

Let $p, q$ be given positive numbers and $a, b$ non-negative ones. In this paper we study and characterize the class $\mathcal S(a,b,p,q)$ of all admissible ideals $\mathcal I \subset 2^\mathbb N$ with the following property $$\sum_{n \in \mathbb N} n^a a^p_n \lt \infty\quad \Rightarrow \quad \mathcal I − \lim n^ba^q_n = 0,$$ for all sequences $(a_n)$ of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic.