Abstract
AbstractA new concept of lacunary statistical boundedness is introduced. It is shown that, for a given lacunary sequenceθ={kr}, a sequence{xk}is lacunary statistical bounded if and only if for ‘almost allkw.r.t.θ’, the valuesxkcoincide with those of a bounded sequence. Apart from studying various algebraic properties and computing the Köthe-Toeplitz duals of the spaceSθ(b)of all lacunary statistical bounded sequences, a decomposition theorem is also established. We characterize thoseθfor whichSθ(b)=S(b). Finally, we give a general description of inclusion between two arbitrary lacunary methods of statistical boundedness.MSC:40C05, 40A05, 46A45.
Highlights
1 Introduction and background Statistical convergence is a generalization of the usual notion of convergence
The idea of statistical convergence was given in the first edition of the monograph of Zygmund [ ], who called it ‘almost convergence’
Statistical convergence arises as an example of ‘convergence in density’ as introduced by Buck [ ]
Summary
Ir |akxk| = for infinitely many r and so k |akxk| = ∞. ] that for a monotone sequence space, α- and β-dual spaces coincide, we have [Sθ (b)]β = [Sθ (b)]α = φ. Proof As [Sθ (b)]αα = φα = ω, so Sθ (b) is not a perfect space. For any lacunary sequence θ , S(b) ⊂ Sθ (b) if and only if lim infr qr >. If lim infr qr > , there exists δ > such that qr ≥ + δ for sufficiently large r
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