ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 40 Badiozzaman, A. J., University of Chamran, Iran On a Tauberian Theorem of G. M. Petersen for Abel summability Let E = \{\underline{s} = (s_{n})_{n≥0} : s_{n} − s_{n−1} = o(n^{−1})\}, where \underline{s} is a sequence of complex numbers. Using the sup norm topology, G. M. Petersen in 1979 defined a type of closure H for E and proved that membership in H is a Tauberian condition for Abel summability. Since E contains unbounded sequences, we define a new norm on E, and using the FK-topology on E+l^{∞}, we show that H = Ē^{E+l^{∞}} and give an alternative proof of Petersen’s result. Also, we give some results for Ω(r), the set of all oscillating sequences. E. g., membership in Ω(r) is a Tauberian condition for Abel summability if and only if there exists a positive integer k such that lim_{n→∞} inf \frac{r_{n+k}}{r_{n}} > 1. References: A. J. Badiozzaman and B. Thorpe, Some best possible Tauberian results for Abel and Cesaro summability II, J. London Math. Soc. (2) 53 (1996), 529–538.