Abstract. We consider continuity theorems for multiplicative integral transforms $\hat{f}$ of functions $f\colon(0, \infty ) \to{\msbm R},$ i.e., we relate the limit behaviours $f_n \to f$ with $\hat{f_n} \to\hat {f}.$ These continuity theorems based on sequential compactness arguments lead to Abelian-- and ratio--Tauberian theorems for these integral transforms.

AMS Subject Classification (1991): 44A05, 40E05

Keyword(s): Integral transform, continuity theorems, Abelian theorems, Tauberian theorems, dominated variation, regular variation

Received January 29, 1999. (Registered under 2707/2009.)