Abstract

For a sequence-to-sequence transformation A, let RmAx=∑n≥m|(Ax)n| and μmAx=supn≥m|(Ax)n|. The purpose of this paper is to study the relationship between the asymptotic equivalence of two sequences (limnxn/yn=1) and the variations of asymptotic equivalence based on the ratios RmAx/RmAy and μmAx/μmAy.