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.

Fundamental properties of statistical convergence and lacunary statistical convergence on time scales

In this paper, we first obtain a Tauberian condition for statistical convergence on time scales. We also find necessary and sufficient conditions for the equivalence of statistical convergence and lacunary statistical convergence on time scales. Some significant applications are also presented.

Generalized Almost Convergence and Core Theorems of Double Sequences

The idea of[λ, μ]-almost convergence (briefly,F[λ, μ]-convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm onF[λ, μ]such that it is a Banach space and then we define and characterize those four-dimensional matrices which transformF[λ, μ]-convergence of double sequencesx=(xjk)intoF[λ, μ]-convergence. We also define aF[λ, μ]-core ofx=(xjk)and determine a Tauberian condition for core inclusions and core equivalence.

On the Slowly Decreasing Sequences of Fuzzy Numbers

We introduce the slowly decreasing condition for sequences of fuzzy numbers. We prove that this is a Tauberian condition for the statistical convergence and the Cesáro convergence of a sequence of fuzzy numbers.

Tauberian conditions for a general limitable method

Let(un)be a sequence of real numbers,Lan additive limitable method with some property, andanddifferent spaces of sequences related to each other. We prove that if the classical control modulo of the oscillatory behavior of(un)inis a Tauberian condition forL, then the general control modulo of the oscillatory behavior of integer ordermof(un)inoris also a Tauberian condition forL.

On a theorem of W. Meyer-König and H. Tietz

Let(un)be a sequence of real numbers and letLbe an additive limitable method with some property. We prove that if the classical control modulo of the oscillatory behavior of(un)belonging to some class of sequences is a Tauberian condition forL, then convergence or subsequential convergence of(un)out ofLis recovered depending on the conditions on the general control modulo of the oscillatory behavior of different order.

A General Tauberian Condition that Implies Euler Summability

Let V be any summability method (whether linear or conservative or not), 0 < p < 1 and s a real or complex sequence. Let Ep denote the matrix of the Euler method. A theorem is proved, giving a condition under which the V-summability of Eps will imply the Ep-summability of s. This extends, in generalized form, an earlier result of N. H. Bingham who considered the case where s is a real sequence and V = B (Borel's method). It is also proved that even for real sequences, the condition given in the theorem cannot be replaced by the condition used by Bingham.