Double sequences and Orlicz functions
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DOUBLE SEQUENCES AND ORLICZ FUNCTIONS T. Yurdakadim1 and E. Tas2 1
Department of Mathematics Faculty of Science, Ankara University Tando˘ gan 06100, Ankara, Turkey E-mail: [email protected]
2
Department of Mathematics Faculty of Science, Ankara University Tando˘ gan 06100, Ankara, Turkey E-mail: [email protected] (Received June 9, 2011; Accepted November 16, 2011) [Communicated by D´enes Petz]
Abstract In this paper we define strong A-convergence with respect to an Orlicz function for double sequences. We show, on bounded double sequences, that statistical convergence and strong A-convergence with respect to any Orlicz function are equivalent. This eliminates a condition of Demirci for bounded single (ordinary) sequences.
1. Introduction In [4] Lindenstrauss and Tzafriri constructed new sequence spaces by using Orlicz functions. Then in [9] Parashar and Choudhary examined these sequence spaces defined by Orlicz functions especially for the Cesaro matrix. The ideas in [1], [9] have been extended for a nonnegative regular matrix and an Orlicz function by Demirci [2]. Demirci showed that stA ∩ l∞ = w(A, F ) ∩ l∞ for an Orlicz function which satisfies the Δ2 -condition. We show that the Δ2 -condition is superfluous. First of all, let us recall preliminary definitions and notations. Definition 1. (Pringsheim 1900) x = (xjk ) is convergent in Pringsheim’s sense (P -convergent) if for every ε > 0 there exists N ∈ N such that |xjk − L| < ε whenever j, k ≥ N ([10]).
Mathematics subject classification numbers: 40F05, 40B05. Key words and phrases: double sequences, Orlicz functions, statistical convergence, strong A-summability. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
T. YURDAKADIM and E. TAS
Definition 2. x = (xjk ) is bounded if there exists a positive number M such that |xjk | < ε for all (j, k) ∈ N2 = N × N, i.e., if x(∞,2) = supj,k |xjk | < ∞. 2 = {x = (xjk ) : x(∞,2) < ∞}. Let l∞ We note that a P -convergent double sequence need not be bounded. In order to see this it is enough to define a double sequence x = (xjk ) by ⎧ ⎨ j, k = 1, xjk = k, j = 1, ⎩ 0, otherwise.
Definition 3. A double sequence x = (xjk ) is said to be statistically convergent to the number L if for each ε > 0 1 |{(j, k) : j ≤ n, k ≤ m, |xjk − L| ≥ ε}| = 0. P-lim nm nm In this case we write st2 -lim x = L. Here the vertical bars denote the cardinality of the enclosed set ([6], [7]). Definition 4. A 4-dimensional matrix A is said to be RH-regular if it maps every bounded P -convergent sequence into a P -convergent sequence with the same P-limit. An RH-regular matrix is characterized as follows. Theorem 1. (Hamilton [3] , Robison [11]) A 4-dimensional matrix A is RH-regular if and only if RH1 : P-limnm anmjk = 0 for each (j, k) ∈ N2 ; RH2 : P-limnm j,k |anmjk | = 1; RH3 : P-limnm j |anmjk | = 0 for each k; RH4 : P-limnm k |anmjk | = 0 for each j; RH5 : j,k anmjk is P -convergent; RH6 : there exist finite positive integers A and B such that j,k>B anmjk < A for
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