# Matrix characterization of oscillation for double sequences

Richard Patterson; Jeff Connor; Jeannette Kline

Open Mathematics (2008)

- Volume: 6, Issue: 3, page 488-496
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topRichard Patterson, Jeff Connor, and Jeannette Kline. "Matrix characterization of oscillation for double sequences." Open Mathematics 6.3 (2008): 488-496. <http://eudml.org/doc/269522>.

@article{RichardPatterson2008,

abstract = {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.},

author = {Richard Patterson, Jeff Connor, Jeannette Kline},

journal = {Open Mathematics},

keywords = {double sequence; p-convergent; oscillation, double oscillations; -convergent; double oscillations},

language = {eng},

number = {3},

pages = {488-496},

title = {Matrix characterization of oscillation for double sequences},

url = {http://eudml.org/doc/269522},

volume = {6},

year = {2008},

}

TY - JOUR

AU - Richard Patterson

AU - Jeff Connor

AU - Jeannette Kline

TI - Matrix characterization of oscillation for double sequences

JO - Open Mathematics

PY - 2008

VL - 6

IS - 3

SP - 488

EP - 496

AB - 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.

LA - eng

KW - double sequence; p-convergent; oscillation, double oscillations; -convergent; double oscillations

UR - http://eudml.org/doc/269522

ER -

## References

top- [1] Agnew R.P., The effects of general regular transformations on oscillations of sequences of functions, Trans. Amer. Math. Soc., 1931, 33, 411–424 http://dx.doi.org/10.2307/1989412 Zbl57.0272.01
- [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] Hurwitz W.A., The oscillation of a sequence, Amer. J. Math., 1930, 52, 611–616 http://dx.doi.org/10.2307/2370629 Zbl56.0201.01
- [4] Mursaleen, Edely O.H., Statistical convergence of double sequences, J. Math. Anal. Appl., 2003, 288, 223–231 http://dx.doi.org/10.1016/j.jmaa.2003.08.004 Zbl1032.40001
- [5] Patterson R.F., Analogues of some fundamental theorems of summability theory, Int. J. Math. Math. Sci., 2000, 23, 1–9 http://dx.doi.org/10.1155/S0161171200001782 Zbl0954.40005
- [6] Pringsheim A., Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 1900, 53, 289–321 (in German) http://dx.doi.org/10.1007/BF01448977
- [7] 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
- [8] Savas E., On some new double sequence spaces defined by a modulus, Appl. Math. Comput, 2007, 187, 417–424 http://dx.doi.org/10.1016/j.amc.2006.08.141 Zbl1126.46300