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.

Tauberian Theorems for Integrals

When, for the generalized summation of series, we use A and B methods, giving A and B sums, respectively, we say that the A method is included in the B method, A ⊂ B, if the B sum exists and is equal to the A sum whenever the latter exists. A theorem proving such a result is called an Abelian theorem. For example, there is an Abelian theorem stating that if the A and B sums are the first Cesàro mean and the Abel mean, respectively, then A ⊂ B. If A ⊂ B and B ⊂ A, we say that A and B are equivalent, A = B. For example, the nth Hölder and nth. Cesàro means are equivalent.