Abel extensions of some classical Tauberian theorems

The well-known classical Tauberian theorems given for Aλ (the discrete Abel mean) by Armitage and Maddox in [Armitage, H. D and Maddox, J. I., Discrete Abel means, Analysis, 10 (1990), 177–186] is generalized. Similarly the ”one-sided” Tauberian theorems of Landau and Schmidt for the Abel method are extended by replacing lim As with Abel-lim Aσi n(s). Slowly oscillating of {sn} is a Tauberian condition of the Hardy-Littlewood Tauberian theorem for Borel summability which is also given by replacing limt(Bs)t = `, where t is a continuous parameter, with limn(Bs)n = `, and further replacing it by Abel-lim(Bσi k (s))n = `, where B is the Borel matrix method.

Equivalence results for discrete Abel means

We present theorems showing when the discrete Abel mean and the Abel summability method are equivalent for bounded sequences and when two discrete Abel means are equivalent for bounded sequences.

Sur la Convergence Ponctuelle de Quelques Suites D'Operateurs

Let (αn.k) be a sequence of positive numbers. We define a regular sequence (resp. a weakly regular sequence) and then show the existence of a unitary operator (resp. a contraction T) L2[0, 1] → L2[0, 1] and a function f ∊ L2[0, 1] such that the pointwise convergence of the sequence of functions is not satisfied almost surely. As a first corollary the pointwise convergence of the Abel means of a contraction from L2 into L2 does not hold necessarily almost surely. As a second corollary there exists a contraction T for which the means (and powers) of Brunei's operator A do not converge pointwise a.s. We also show that, for P > 1 fixed, there exists a sequence of positive numbers αn.k for which we have the pointwise convergence in LP of the sequence of polynomials where T is a contraction of L1 and Lα. The dominated theorem does not, however, always hold for such LP-contractions.