Extension of Maddox’s space l(μ) with Nörlund means

In this paper, we introduce a new series space [Formula: see text] as the set of all series summable by the absolute Nörlund summability method, which includes the spaces [Formula: see text] and [Formula: see text] of Maddox [Spaces of strongly summable sequences, Quart. J. Math. 18(1) (1967) 345–355], Sarıgöl [Spaces of series summable by absolute Cesàro and matrix operators, Comm. Math Appl. 7(1) (2016) 11–22], Hazar and Sarıgöl [On absolute Nörlund spaces and matrix operators, Acta Math. Sinica, (English Ser.) 34(5) (2018) 812–826], respectively. Also, we study its some algebraic and topological structures such as isomorphism, the [Formula: see text], [Formula: see text], [Formula: see text] duals, Schauder basis, and characterize certain matrix transformations on that space.

Inclusion and equivalence relations between absolute Norlund and absolute weighted mean summability methods

In this paper, a set of conditions under which the absolute Nörlund summability method include in the absolute weighted mean method have been established. Three non-trivial examples to show that this inclusion holds have been given, and other three examples to show that even if both and are regular, the inclusion fails to holds have been constructed. The paper give two non-trivial examples to show that the equivalence of these two methods may holds. Finally, we give two examples to show that inclusion may holds in only one way without the other.

Quasimonotone and Almost Increasing Sequences and Their New Applications

Recently, we have proved a main theorem dealing with the absolute Nörlund summability factors of infinite series by using -quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained.

On absolute Norlund summability of Fourier series

We obtain two theorems on the absolute Norlund summability of Fourier series and factored Fourier series.

ON ABSOLUTE NORLUND SUMMABILITY OF ORTHOGONAL SERIES

The purpose of this paper is to give a general theorem on the $|N_{,p_n};\delta|_k$ summability of orthogonal series, which generalizes a theorem due to Okuyama [1] related to summability of orthogonal series.

Some absolutely effective product methods

It is proved that the product methodA(C,1), where(C,1)is the Cesàro arithmetic mean matrix, is totally effective under certain conditions concerning the matrixA. This general result is applied to study absolute Nörlund summability of Fourier series and other related series.