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• # Artykuł - szczegóły

## Open Mathematics

2008 | 6 | 4 | 581-594

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

EN

### Abstrakty

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.

EN

581-594

wydano
2008-12-01
online
2008-10-08

### Twórcy

autor
• University of North Florida
autor
• Savannah State University

### Bibliografia

• [1] Agnew R.P., On oscillations of real sequences and of their transforms by square matrices, Amer. J. Math., 1939, 61, 683–699 http://dx.doi.org/10.2307/2371323
• [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
• [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