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THE v-INVARIANT χ2 SEQUENCE SPACES

Year 2011, Volume: 01 Issue: 2, 173 - 184, 01.12.2011

Abstract

In this paper we define v− invariatness of a double sequence space of χ and examine the v− invariatness of the double sequence space of χ. Furthermore, we give duals of double sequence space of χ.

References

  • [1] Aposto, T., (1978), Mathematical Analysis, Addison-Wesley, London.
  • [2] Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
  • [3] Bektas, C. and Altin, Y., (2003), The sequence space `M (p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
  • [4] Bromwich, T.J.I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
  • [5] Hardy, G.H., (1917), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19, 86-95.
  • [6] Krasnoselskii, M.A. and Rutickii, Y.B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
  • [7] Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
  • [8] Maddox, I.J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
  • [9] Moricz, F., (1991), Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
  • [10] Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
  • [11] Mursaleen, M., Khan, M.A. and Qamaruddin, (1999), Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
  • [12] Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
  • [13] Orlicz, W., (1936), U¨ber Raume ` L M´ Bull. Int. Acad. Polon. Sci. A, 93-107.
  • [14] Parashar, S.D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(4), 419-428.
  • [15] Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
  • [16] Ruckle, W.H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
  • [17] Tripathy, B.C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
  • [18] Tripathy, B.C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
  • [19] Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
  • [20] Kamthan, P.K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
  • [21] G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces c P 2 (p) and c P B 2 (p), Appl. Math. Comput., 157(2), 491-501.
  • [22] G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces ` ∞2 , ibid., 160(1), 147-153.
  • [23] Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
  • [24] Mursaleen, M. and Edely, O.H.H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
  • [25] Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
  • [26] Mursaleen, M. and Edely, O.H.H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
  • [27] Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
  • [28] Basar, F. and Y.Sever, Y., (2009), The space Lp of double sequences, Math. J. Okayama Univ, 51, 149-157.
  • [29] Subramanian, N. and Misra, U.K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
  • [30] Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
  • [31] Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
  • [32] Maddox, I.J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
  • [33] Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2), 194-198.
  • [34] Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
  • [35] Hamilton, H.J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
  • [36] ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
  • [37] ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
  • [38] ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
  • [39] , Robison, G.M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
  • [40] Silverman, L.L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
  • [41] Toeplitz, O., (1911), U¨ber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (warsaw), 22.
  • [42] Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
  • [43] Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p,(0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
  • [44] C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.
Year 2011, Volume: 01 Issue: 2, 173 - 184, 01.12.2011

Abstract

References

  • [1] Aposto, T., (1978), Mathematical Analysis, Addison-Wesley, London.
  • [2] Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
  • [3] Bektas, C. and Altin, Y., (2003), The sequence space `M (p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
  • [4] Bromwich, T.J.I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
  • [5] Hardy, G.H., (1917), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19, 86-95.
  • [6] Krasnoselskii, M.A. and Rutickii, Y.B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
  • [7] Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
  • [8] Maddox, I.J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
  • [9] Moricz, F., (1991), Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
  • [10] Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
  • [11] Mursaleen, M., Khan, M.A. and Qamaruddin, (1999), Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
  • [12] Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
  • [13] Orlicz, W., (1936), U¨ber Raume ` L M´ Bull. Int. Acad. Polon. Sci. A, 93-107.
  • [14] Parashar, S.D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(4), 419-428.
  • [15] Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
  • [16] Ruckle, W.H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
  • [17] Tripathy, B.C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
  • [18] Tripathy, B.C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
  • [19] Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
  • [20] Kamthan, P.K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
  • [21] G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces c P 2 (p) and c P B 2 (p), Appl. Math. Comput., 157(2), 491-501.
  • [22] G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces ` ∞2 , ibid., 160(1), 147-153.
  • [23] Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
  • [24] Mursaleen, M. and Edely, O.H.H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
  • [25] Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
  • [26] Mursaleen, M. and Edely, O.H.H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
  • [27] Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
  • [28] Basar, F. and Y.Sever, Y., (2009), The space Lp of double sequences, Math. J. Okayama Univ, 51, 149-157.
  • [29] Subramanian, N. and Misra, U.K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
  • [30] Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
  • [31] Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
  • [32] Maddox, I.J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
  • [33] Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2), 194-198.
  • [34] Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
  • [35] Hamilton, H.J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
  • [36] ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
  • [37] ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
  • [38] ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
  • [39] , Robison, G.M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
  • [40] Silverman, L.L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
  • [41] Toeplitz, O., (1911), U¨ber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (warsaw), 22.
  • [42] Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
  • [43] Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p,(0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
  • [44] C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.
There are 44 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

N. Subramanian This is me

U. K. Misra This is me

Publication Date December 1, 2011
Published in Issue Year 2011 Volume: 01 Issue: 2

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