On the paranormed Nörlund difference sequence space of fractional order and geometric properties

In this article we introduce paranormed Nörlund difference sequence space of fractional order α, Nt (p, Δ(α)) defined by the composition of fractional difference operator Δ(α), defined by ( Δ ( α ) x ) k = ∑ i = 0 ∞ ( − 1 ) i Γ ( α + 1 ) i ! Γ ( α − i + 1 ) x k − i , $\begin{array}{} \displaystyle (\Delta^{(\alpha)}x)_k=\sum_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i}, \end{array}$ and Nörlund matrix Nt . We give some topological properties, obtain the Schauder basis and determine the α−, β− and γ-duals of the new space. We characterize certain matrix classes related to this new space. Finally we investigate certain geometric properties of the space Nt (p, Δ(α)).

On Domain of Nörlund Sequences

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α­, β­, γ­duals, and characterized their matrix transformations on this space and into this space.

Nörlund Operators On lp

The Nörlund matrix Na is the triangular matrix {an-k /An}, where an ≥ 0 and An := a0 + a1 + • • • + an > 0. It is proved that, subject to the existence of α := lim nan/An, Na ∊ B(lp) for 1 < p < ∞ if and only if α < ∞. It is also proved that it is possible to have Na ∊ B(lp) for 1 < p < ∞ when sup nan/An = ∞.

Absolute Nörlund matrix summability of Fourier series based on inclusion theorems

In a recent paper [39] the second author established sufficient conditions for a Nörlund matrix (N, p) to be stronger than a Cesaro matrix (C, α) of order α, for some positive α. This result was then used to show that a number of known theorems dealing with the Nörlund summability of Fourier series, or related series, can be more easily established, since the conditions placed on the Nörlund matrix imply that it is stronger than some (C, α) method, for which the summability theorem has already been established.