Some inclusion theorems for absolute summability

In this work the inclusion relations between absolute summability domains of a normal matrix A and certain factorable matrices are described. Thus, some classes of factorable matrices transforming the absolute summability domain of A into a set of convergent or absolutely convergent series are characterized. As an application, the special case where A is the Cesàro matrix is considered.     

Some equivalence theorems on absolute summability methods

We obtained necessary and sufficient conditions for the equivalence of two general summability methods. Some new and known results are also obtained.     

Some Tauberian theorems for Borel summability methods

We investigate the conditions needed for a Borel summable sequence to be convergent. The results of this paper extend and improve the well-known result of Hardy and Littlewood (1913) [1].     

General absolute summability factor theorems involving quasi-power-increasing sequences

In this paper a general theorem concerning the |A,δ|_k summability methods has been proved, which generalizes two results of Şevli and Leindler [H. Şevli and L. Leindler, On the absolute summability factors of infinite series involving quasi-power-increasing sequences, Computers and Mathematics with Applications 57 (2009), 702–709]. We obtain sufficient conditions for ∑a_nλ_n to be summable |A,δ|_k, k1, 0δ<1/k, by using quasi-power-increasing sequences.     

Some relations between Nörlund and Abel summability

The definition of generalised Abel summability is extended to positive orders and a definition of strong generalised Abel summability is introduced. A result of Jurkat and Peyerimhoff concerning the implication between Nörlund and generalised Abel summabilities is extended to positive integral order. Analogous results for the corresponding strong methods are also given.

     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

Extension on absolute summability factors of infinite series

Extension and improvement has been made on a recent result proved by Mazhar [S.M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999) 67-73] concerning absolute summability factors of infinite series.     

Tauberian theorems for the logarithmic summability methods of integrals

Positivity - A real-valued continuous function f on $$[1, infty )$$ is said to be summable by the logarithmic summability method of integrals (shortly, $$ell $$ summable) if $$begin{aligned}...     

A note on two absolute summability methods

In this paper we have established a relation between |R,p _n ;δ|_k and |R,q _n ;δ|_k summability methods which generalizes the results of Bor [1] and Bosanquet [2].     

On generalized absolute Cesàro summability

In this paper, a known result on ∣C, α; δ∣_k summability factors has been generalized for ∣C, α, γ; δ∣_k summability factors. Some new results have also been obtained.     

On (N, p, q) summability factors of infinite series

In this paper a necessary and sufficient condition has been obtained for Σa_n∈_n to be summable |N, q| whenever Σa_n is bounded (N, p, q).     

Some remarks on general commutators theorems

The paper proves some general facts on commutators that refer to Fuglede-Putnam classical theorem in the spectral theory of not necessarily selfadjoint operators.     

Almost everywhere summability of multiple Walsh-Fourier series

We prove that for the N-dimensional Walsh-Fourier series the maximal operator of the Marcinkiewicz means is of weak type (1,1). Moreover, the Marcinkiewicz means σ_nf of the function f∈L¹ converge a.e. to f as n→∞.     

A general theorem characterizing some absolute summability methods

A general theorem is given which gives the necessary and sufficient conditions satisfied by a sequence (ε_n) in order to have the series Σa _n ε _n summable to |A| whenever Σa _n is summable to |A| for some summability methodA.