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

Richard Patterson; Mulatu Lemma

Open Mathematics, Tome 6 (2008), p. 581-594 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $P - \underset{(m, n) \to \infty; (α, β) \to \infty}{l i m s u p} |σ_{m, n} - σ_{α, β}| ⩽ P - \underset{(m, n) \to \infty; (α, β) \to \infty}{l i m s u p} |s_{m, n} - s_{α, β}|$ for each double real bounded sequences s k;l where $σ_{m, n} = \sum_{k, l = 1, 1}^{\infty, \infty} a_{m, n, k, l,} s_{k, l} .$ In addition, other implications and variations are also presented.

Publié le : 2008-01-01

Zbl 1165.40004

EUDML-ID : urn:eudml:doc:269368

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     author = {Richard Patterson and Mulatu Lemma},
     title = {Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices},
     journal = {Open Mathematics},
     volume = {6},
     year = {2008},
     pages = {581-594},
     zbl = {1165.40004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0043-7}
}

Richard Patterson; Mulatu Lemma. Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices. Open Mathematics, Tome 6 (2008) pp. 581-594. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0043-7/

Bibliographie

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[2] Hamilton H.J., Transformations of multiple sequences, Duke Math. J., 1936, 2, 29–60 http://dx.doi.org/10.1215/S0012-7094-36-00204-1 | Zbl 0013.30301

[3] Pringsheim A., Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 1900, 53, 289–321 (in German) http://dx.doi.org/10.1007/BF01448977

[4] Robison G.M., Divergent double sequences and series, Trans. Amer. Math. Soc., 1926, 28, 50–73 http://dx.doi.org/10.2307/1989172 | Zbl 52.0223.01