Almost Regular Matrices

  • M. Mursaleen
  • S. A. Mohiuddine

Abstract

The Silverman–Toeplitz theorem is a well-known theorem that states necessary and sufficient conditions to transform a convergent sequence into a convergent sequence leaving the limit invariant. This idea was extended to RH-regular matrices by using the notion of P-convergence (see Hamilton in Duke Math. J. 2:29–60, 1936 and Robinson in Trans. Am. Math. Soc. 28:50–73, 1926). In this chapter, we use the notion of almost convergence to define and characterize almost conservative, almost regular, strongly regular, and almost strongly regular four-dimensional matrices.

Keywords

Four-dimensional infinite matrices \(\mathcal{C}_{\nu }\)-conservative and \(\mathcal{C}_{\nu }\)-regular matrices Bounded-regular matrices Almost \(\mathcal{C}_{\nu }\)-conservative and almost \(\mathcal{C}_{\nu }\)-regular matrices Strongly regular matrices Almost strongly regular 

References

  1. 39.
    J.P. Duran, Infinite matrices and almost convergence. Math. Z. 128, 75–83 (1972) MathSciNetCrossRefMATHGoogle Scholar
  2. 50.
    H.J. Hamilton, Transformations of multiple sequences. Duke Math. J. 2, 29–60 (1936) MathSciNetCrossRefGoogle Scholar
  3. 55.
    J.P. King, Almost summable sequences. Proc. Am. Math. Soc. 17, 1219–1225 (1966) CrossRefMATHGoogle Scholar
  4. 63.
    G.G. Lorentz, A contribution to theory of divergent sequences. Acta Math. 80, 167–190 (1948) MathSciNetCrossRefMATHGoogle Scholar
  5. 83.
    F. Móricz, B.E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Camb. Philos. Soc. 104, 283–294 (1988) CrossRefMATHGoogle Scholar
  6. 88.
    M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences. J. Math. Anal. Appl. 293, 523–531 (2004) MathSciNetCrossRefMATHGoogle Scholar
  7. 103.
    M. Mursaleen, E. Savaş, Almost regular matrices for double sequences. Studia Sci. Math. Hung. 40, 205–212 (2003) CrossRefMATHGoogle Scholar
  8. 111.
    G.M. Robinson, Divergent double sequences and series. Trans. Am. Math. Soc. 28, 50–73 (1926) CrossRefGoogle Scholar
  9. 128.
    M. Zeltser, M. Mursaleen, S.A. Mohiuddine, On almost conservative matrix methods for double sequence spaces. Publ. Math. (Debr.) 75, 387–399 (2009) MathSciNetMATHGoogle Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  • M. Mursaleen
    • 1
  • S. A. Mohiuddine
    • 2
  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

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