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Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices

100%
Patterson R., Lemma M.
Open Mathematics
|
2008
|
tom 6
|
nr 4
581-594
EN
In 1939 Agnew presented a series of conditions that characterized the oscillation of ordinary sequences using ordinary square conservative matrices and square multiplicative matrices. The goal of this paper is to present multidimensional analogues of Agnew’s results. To accomplish this goal we begin by presenting a notion for double oscillating sequences. Using this notion along with square RH-conservative matrices and square RH-multiplicative matrices, we will present a series of characterization of this sequence space, i.e. we will present several necessary and sufficient conditions that assure us that a square RH-multiplicative(square RH-conservative) be such that $$ P - \mathop {limsup}\limits_{(m,n) \to \infty ;(\alpha ,\beta ) \to \infty } \left| {\sigma _{m,n} - \sigma _{\alpha ,\beta } } \right| \leqslant P - \mathop {limsup}\limits_{(m,n) \to \infty ;(\alpha ,\beta ) \to \infty } \left| {s_{m,n} - s_{\alpha ,\beta } } \right| $$ for each double real bounded sequences {s k;l} where $$ \sigma _{m,n} = \sum\limits_{k,l = 1,1}^{\infty ,\infty } {a_{m,n,k,l,} s_{k,l} } . $$ In addition, other implications and variations are also presented.
2
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Matrix characterization of oscillation for double sequences

100%
Patterson R., Connor J., Kline J.
Open Mathematics
|
2008
|
tom 6
|
nr 3
488-496
EN
The notion of oscillation for ordinary sequences was presented by Hurwitz in 1930. Using this notion Agnew and Hurwitz presented regular matrix characterization of the resulting sequence space. The primary goal of this article is to extend this definition to double sequences, which grants us the following definition: the double oscillation of a double sequence of real or complex number is given P-lim sup(m,n)→∞;(α,β)→∞|S m,n-S α,β|. Using this concept a matrix characterization of double oscillation sequence space is presented. Other implication and variation shall also be presented.
3
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Extended Real-Valued Double Sequence and Its Convergence

100%
Endou N.
Formalized Mathematics
|
2015
|
tom 23
|
nr 3
253-277
EN
In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.
4
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Tauberian theorems for Cesàro summable double sequences

75%
Móricz F.
Studia Mathematica
|
1994
|
tom 110
|
nr 1
83-96
EN
$(s_{jk}: j,k = 0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_{jk})$ converges in Pringsheim's sense. These conditions are satisfied if $(s_{jk})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_{jk})$ is summable (C,1,1) to a finite limit and there exist constants $n_1 > 0$ and H such that $jk(s_{jk} - s_{j-1,k} - s_{j-1,k} + s_{j-1,k-1}) ≥ -H$, $j(s_{jk} - s_{j-1, k}) ≥ -H$ and $k(s_{jk} - s_{j,k-1}) ≥ -H$ whenever $j,k > n_1$, then $(s_{jk})$ converges. We always mean convergence in Pringsheim's sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.
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