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Strongly almost \((w, \lambda)\)-summable sequences defined by Orlicz functions. (English) Zbl 1099.46004

The present paper combines three topics: strong summability, statistical convergence, and Orlicz functions. Recall that an Orlicz function \(M: [0,\infty)\to [0,\infty)\) is continuous, nondecreasing and convex with \(M(0)= 0\), \(M(x)> 0\) for \(x> 0\) and \(M(x)\to\infty\) as \(x\to\infty\). If \(M\) is subadditive instead of convex, then it is called a modulus function [W. H.Ruckle, Can.J.Math.25, 973–978 (1973; Zbl 0267.46008), I. J.Maddox, Math.Proc.Camb.Philos.Soc.100, 161–166 (1986; Zbl 0631.46010)].
In [Q. J.Math., Oxf.II.Ser.18, 345–355 (1967; Zbl 0156.06602)], I. J.Maddox defined the sets \([A, p]\), \([A, p]_0\) and \([A, p]_\infty\) of strongly summable, strongly summable to \(0\) and strongly bounded sequences, respectively. In his definitions, \(A= (a_{nk})_{n,k\in\mathbb{N}}\) is an infinite matrix of complex numbers and \(p= (p_k)_{k\in\mathbb{N}}\) is a sequence of strictly positive numbers. By specializing the matrix \(A\), he obtained some well-known sequence spaces. He introduced a kind of generalized Cesàro summability \([C,1, p]\) by taking \(A= (C,1)\) and defining that \(x_k\to\ell[C,1, p]\) iff \({1\over n}\sum^n_{k=1} |x_k-\ell|^{p_k}\to 0\). He also defined the spaces \([C,1, p]_0\) and \([C,1,p]_\infty\). The same author gave, in 1978, his definition of a strongly almost convergent sequence: A sequence \(x= (x_k)_{k\in\mathbb{N}}\) is strongly almost convergent to a number \(L\) iff \(\lim_n{1\over n} \sum^n_{k=1} |x_{k+m}- L|= 0\) uniformly in \(m\).
Since then, various authors worked along similar lines and defined sequence spaces which, in certain cases, were expected generalizations of the above definitions. If, for example, we replace the term \(x_{k+m}\) in the previous definition by the quantity \(t_{km}= {x_m+ x_{m+1}+\cdots+ x_{m+k}\over k+1}\), we obtain the space \([w]\) of G. Das and S. K.Sahoo [J.Indian Math.Soc., New Ser.58, No. 1–4, 65–74 (1992; Zbl 0880.40004)].
Let \(\lambda= (\lambda_n)\) be a non-decreasing sequence of positive numbers tending to \(\infty\) with \(\lambda_{n+1}\leq \lambda_n+ 1\), \(\lambda_1= 1\) and set \(I_n= [n- \lambda_n+ 1,n]\). A sequence \(x= (x_k)_{k\in \mathbb{N}}\) is defined to be strongly (\(w,\lambda)\)-summable to \(L\) iff \(\lim_n{1\over \lambda_n}\sum_{k\in I_n} |t_{km}- L|= 0\) uniformly in \(m\). The space of strongly \((w,\lambda)\)-summable sequences is denoted by \([w,\lambda]\). The authors, extending ideas of J. Lindenstrauss and L. Tzafriri [Isr.J.Math.10, 379–390 (1971; Zbl 0227.46042)] and of Maddox, use an Orlicz function \(M\) and a sequence \(p= (p_k)_{k\in\mathbb{N}}\) of strictly positive numbers to create the set of sequences \([w,\lambda, M,p]\), where \(x= (x_k)\in[w, \lambda,M,p]\) iff \(\lim_n{1\over \lambda_n} \sum_{k\in I_n} [M({|t_{km}- L|\over\rho})]^{p_k}= 0\) uniformly in \(m\) for some \(L\) and for some \(\rho> 0\). They also define the sets \([w,\lambda,M, p]_0\), \([w,\lambda, M,p]_\infty\) in a respective manner. In the special case \(p_k=1\;\forall k\), a sequence \(x= (x_k)\in[w,\lambda M]:= [w,\lambda, M,(p_k= 1\;\forall k)]\) is defined to be strongly \((w,\lambda)\)-summable with respect to the Orlicz function \(M\).
After studying some properties of the above sets, the authors prove that every sequence in \([w,\lambda M]\) is \(\overline S_\lambda\)-statistically convergent, i.e., there exists \(L\) such that for every \(\varepsilon> 0\) one has that \(\lim_n {1\over\lambda_n} |\{k\in I_n:|t_{km}- L|\geq \varepsilon\}|= 0\) uniformly in \(m\). Moreover, they prove that if the condition \(M(x)\to\infty\) as \(x\to\infty\) in the definition of Orlicz function is dropped while \(M\) is a bounded function, then \(\overline S_\lambda \subset[w,\lambda,M]\), where \(\overline S_\lambda\) is the set of \(\overline S_\lambda\)-statistically convergent sequences.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40A05 Convergence and divergence of series and sequences
40C05 Matrix methods for summability
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