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Mercerian theorems involving Cesàro means of positive order

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Abstract

IfC α denotes the Cesàro matrix of arbitrary positive order α, we investigate the setD α of complex numbers μ for which the sequence-to-sequence transformation defined by the matrixI−μC α is equivalent to convergence. In particular, we show thatD α is strictly decreasing as α increases. Most previous investigations have taken μ real and/or α an integer; we make no such restrictions.

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References

  1. Agnew, R. P.: Convergence fields of methods of summability. Annals Math.46, 93–101 (1945).

    Google Scholar 

  2. Beekmann, W.: Mercer-Sätze für abschnittsbeschränkte Matrixtransformationen. Math. Z.97, 154–157 (1967).

    Google Scholar 

  3. Jakimovski, A.: Some relations between the methods of summability of Abel, Borel, Cesàro, Hölder and Hausdorff. J. d'Anal. Math.3, 346–381 (1953/54).

    Google Scholar 

  4. Knopp, K.: Theorie und Anwendung der unendlichen Reihen, 5. Aufl. Berlin-Göttingen-Heidelberg-New York: Springer. 1964.

    Google Scholar 

  5. Kuttner, B.: Mercerian theorems involving Cesàro or Hölder means of any positive order. J. Indian Math. Soc.34, 151–158 (1970).

    Google Scholar 

  6. Love, E. R.: Mercer's summability theorem. J. London Math. Soc.27, 413–428 (1952).

    Google Scholar 

  7. Nehari, Z.: Conformal Mapping. New York: McGraw-Hill. 1952.

    Google Scholar 

  8. Pitt, H. R.: Mercerian theorems. Proc. Camb. Phil. Soc.34, 510–520 (1938).

    Google Scholar 

  9. Polniakowski, Z.: Polynomial Hausdorff transformations, I. Mercerian theorems. Ann. Polonici Math.5, 1–24 (1958).

    Google Scholar 

  10. Szüsz, P.: On Mercer's theorem for (C, 2)-means. Mh. Math.81, 149–152 (1976).

    Google Scholar 

  11. Whittaker, E. T., Watson, G. N.: Modern Analysis. 4th Ed. Cambridge: Univ. Press. 1963.

    Google Scholar 

  12. Wilansky, A.: Functional Analysis. New York: Blaisdell. 1964.

    Google Scholar 

  13. Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin-Heidelberg-New York: Springer. 1970.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel

    A. Jakimovski

  2. Department of Mathematics, York University, M3J 1P3, Toronto, Canada

    D. C. Russell

Authors
  1. A. Jakimovski
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  2. D. C. Russell
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Jakimovski, A., Russell, D.C. Mercerian theorems involving Cesàro means of positive order. Monatshefte für Mathematik 96, 119–131 (1983). https://doi.org/10.1007/BF01530688

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