Publ. Math. Debrecen 61 / 1-2 (2002), 1–9 Some Tauberian theorems for Schwartz distributions By RICARDO ESTRADA (San Jos´ e) Abstract. We give a condition on a space of test functions A for the inclusion A∩A ′ ⊂S to hold. These kinds of results are Tauberian theorems that guarantee that a generalized function of rapid distributional decay at infinity is a rapidly decreasing smooth function, that is, an element of S. 1. Introduction The purpose of this article is to study some class of Tauberian theo- rems for generalized functions. Our main concern is to study some supple- mentary conditions on a smooth generalized function that decays distribu- tionally at infinity which guarantee that the function is a rapidly decreasing smooth function, that is, an element of the space of test functions S . More specifically, we identify a condition on the space of test functions A that guarantees that (1.1) A∩A ′ ⊂S . Results of this kind are very useful in mathematical physics [4, Section 4]. We mention the results O M ∩O ′ M ⊂S and O C ∩O ′ C ⊂S given by Ortner and Wagner [12] and later in [4]. Observe the Tauberian character of such a result: the elements of O ′ M are generalized functions that in some sense (made precise in Section 2) decay at infinity; the elements of O M are smooth functions that do not increase too fast at infinity, but which are usually not of rapid decay. The fact that O M ∩O ′ M ⊂S is thus a rather interesting result. Mathematics Subject Classification : 46F10. Key words and phrases : Tauberian theorem, distributional decay.