Ideal extensions of Olivier's theorem (Q2054566)

The authors study and characterize the class \(S(a, b, p, q) \), \(p,q >0\), \(a,b \geq 0\) of all admissible ideals \(I \subset 2^{\mathbb N}\) with the property \[ \sum_{n \in \mathbb N}n^{a} a_{n}^{p} < \infty\Longrightarrow I-\lim n^{b} a_{n}^{q} = 0 \] for all sequences \((a_n)\) of real numbers. Additionally, in some corollaries they present special cases including some previously published theorems on this topic.