Báo cáo toán học: Lacunary Strongly Summable Sequences and q-Lacunary Almost Statistical Convergence

Một chuỗi lacunary là một chuỗi ngày càng tăng θ = (kr) số nguyên dương như vậy mà k0 = 0 và kr-kr-1 → ∞ như r → ∞. Một chuỗi x = (xk) được gọi là q-lacunary gần như hội tụ thống kê để ξ cung cấp cho mỗi ε0, limr (kr-kr-1) -1 {số k: kr-1...
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Báo cáo toán học: "Lacunary Strongly Summable Sequences and q-Lacunary Almost Statistical Convergence"Vietnam Journal of Mathematics 34:2 (2006) 129–138 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 Lacunary Strongly Summable Sequences and q-Lacunary Almost Statistical Convergence Rifat Colak1 , B. C. Tripathy2 , and Mikˆil Et1 ¸ a 1 Department of Mathematics, Firat University, 23119, Elazı˘-Turkey g 2 Mathematical Sciences Division Institute of Advanced Study in Science and Technology, Paschim Baragoan, Garchuk, Guwahati 781035, Assam, India Received January 28, 2005 Revised February 28, 2006Abstract. A lacunary sequence is an increasing sequence θ=(kr ) of positive integerssuch that k0 =0 and kr −kr−1 →∞ as r→∞. A sequence x=(xk ) is called q−lacunary almoststatistical convergent to ξ provided that for each ε>0, limr (kr −kr−1 )−1 { the number ofk:kr−1 130 Rifat Colak, B. C. Tripathy, and Mikˆil Et ¸ ai) L(x) 0 if x 0 (i.e. xn 0 for all n),ii) L(e) = 1, where e = (1, 1, . . . ),iii) L(Dx) = L(x),where D is the shift operator defined by (Dxn ) = (xn+1 ). Let B be the set of all Banach limits on ∞ . A sequence x is said to be almostconvergent to a number ξ if L(x) = ξ for all L ∈B. Lorentz [12] has shown thatx is almost convergent to ξ if and only if xm + xm+1 + . . . + xm+k tkm = tkm (x) = → ξ as k → ∞, uniformly in m. k+1Let f denote the set of all almost convergent sequences. We write f − lim x = ξif x is almost convergent to ξ. Maddox [13] and (independently) Freedman et al.[7] have defined x to be strongly almost convergent to a number ξ if k 1 |xi+m − ξ | → 0 as k → ∞, uniformly in m. k+1 i=0 Let [f ] denote the set of all strongly almost convergent sequences. If x isstrongly almost convergent to ξ, we write [f ] − lim x = ξ. It is easy to see that[f ] ⊂ f ⊂ ∞ . Das and Sahoo [4] defined the sequence space n 1 |tkm (x) − ξ |pk → 0 as n → ∞, uniformly in m [w(p)] = x ∈ w : n+1 k=0and investigated some of its properties. The definition of statistical convergence was introduced by Fast [6] in a shortnote. Schoenberg [20] studied statistical convergence as a summability methodand listed some of the elementary properties of statistical convergence. Recently,statistical convergence has been studied by various authors (cf. [3, 8, 9, 14, 17,18]). The statistical convergence depends on the density of the subsets of N, theset of natural numbers. A subset E of N is said to have density δ (E ) if n 1 δ (E ) = lim χE (k ) exists, n→∞ n k=1where χE is the characteristic function of E. A sequence (xn ) is said to be statistically convergent to ξ if for every ε >0, δ {k ∈ N : |xk − ξ | ε} = 0. In this case we write stat-lim xk = ξ. Let θ = (kr ) be the sequence of positive integers such that k0 = 0, 0 < kr 0 for x > 0 and M (x) → ∞as x → ∞.Lacunary Sequences and Almost Statistical Convergence 131 Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to constructthe sequence space ∞ |xk | = x∈w: M < ∞ for some ρ > 0 . M ρ k=1The space with the norm M ∞ ...