Abel Transformations Into I 1

Fridy, J. A.

Canadian mathematical bulletin, Tome 25 (1982) no. 4, pp. 421-427

Résumé

Let t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by a nk = t n(1-t n )k . The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l 1 into l 1 if and only if t is in l 1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ x k has bounded partial sums, then A t x is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At .

Détail

DOI : 10.4153/CMB-1982-060-5

Mots-clés : 40D05, 40D25, 40E05, 40G10