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

Richard Patterson; Mulatu Lemma

Open Mathematics (2008)

  • Volume: 6, Issue: 4, page 581-594
  • ISSN: 2391-5455

Abstract

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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 - l i m s u p ( m , n ) → ∞ ; ( α , β ) → ∞ σ m , n - σ α , β ⩽ P - l i m s u p ( m , n ) → ∞ ; ( α , β ) → ∞ s m , n - s α , β for each double real bounded sequences s k;l where σ m , n = ∑ k , l = 1 , 1 ∞ , ∞ a m , n , k , l , s k , l . In addition, other implications and variations are also presented.

How to cite

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Richard Patterson, and Mulatu Lemma. "Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices." Open Mathematics 6.4 (2008): 581-594. <http://eudml.org/doc/269368>.

@article{RichardPatterson2008,
abstract = {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 \[ P - \mathop \{limsup\}\limits \_\{(m,n) \rightarrow \infty ;(\alpha ,\beta ) \rightarrow \infty \} \left| \{\sigma \_\{m,n\} - \sigma \_\{\alpha ,\beta \} \} \right| \leqslant P - \mathop \{limsup\}\limits \_\{(m,n) \rightarrow \infty ;(\alpha ,\beta ) \rightarrow \infty \} \left| \{s\_\{m,n\} - s\_\{\alpha ,\beta \} \} \right| \] 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.},
author = {Richard Patterson, Mulatu Lemma},
journal = {Open Mathematics},
keywords = {double sequence; p-convergent; oscillation; double oscillations},
language = {eng},
number = {4},
pages = {581-594},
title = {Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices},
url = {http://eudml.org/doc/269368},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Richard Patterson
AU - Mulatu Lemma
TI - Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices
JO - Open Mathematics
PY - 2008
VL - 6
IS - 4
SP - 581
EP - 594
AB - 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 \[ P - \mathop {limsup}\limits _{(m,n) \rightarrow \infty ;(\alpha ,\beta ) \rightarrow \infty } \left| {\sigma _{m,n} - \sigma _{\alpha ,\beta } } \right| \leqslant P - \mathop {limsup}\limits _{(m,n) \rightarrow \infty ;(\alpha ,\beta ) \rightarrow \infty } \left| {s_{m,n} - s_{\alpha ,\beta } } \right| \] 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.
LA - eng
KW - double sequence; p-convergent; oscillation; double oscillations
UR - http://eudml.org/doc/269368
ER -

References

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  1. [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 Zbl0021.21901
  2. [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 Zbl0013.30301
  3. [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. [4] Robison G.M., Divergent double sequences and series, Trans. Amer. Math. Soc., 1926, 28, 50–73 http://dx.doi.org/10.2307/1989172 Zbl52.0223.01

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