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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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On the Pringsheim convergence of double series; pp. 108–121

(Full article in PDF format) doi: 10.3176/proc.2009.2.03


Authors

Balmohan V. Limaye, Maria Zeltser

Abstract

Several aspects of the convergence of a double series in the sense of Pringsheim are considered in analogy with some well-known results for single series. They include various tests for absolute convergence and also criteria for convergence of the Cauchy product. Some errors in the works of earlier authors are corrected.

Keywords

summability theory, double series, convergence test, Pringsheim convergence, boundedly convergent, regularly convergent, Cauchy product, matrix transformation.

References

  1. Baron , S. B. Derivation of tests for the convergence of double numerical series. Tartu Riikl. Ülik. Toimetised , 1958 , 55 , 9–19 (in Russian).

  2. Biermann , O. Über unendliche Doppelreihen und unendliche Doppelproducte. Monats. Math. Phys. , 1897 , 8 , 115–137.
doi:10.1007/BF01696269

  3. Boos , J. Classical and Modern Methods in Summability. Oxford University Press , Oxford , 2000.

  4. Buck , R. C. Advanced Calculus , 3rd edn. McGraw-Hill , New York , 1978.

  5. Cesari , L. Sulla moltiplicazione delle serie doppie. Ann. Scuola Norm. Sup. Pisa , 1947 , 12 , 189–204.

  6. Ghorpade , S. R. and Limaye , B. V. A Course in Calculus and Real Analysis. Springer , New York , 2006.

  7. Hamilton , H. J. On transformations of double series. Bull. Amer. Math. Soc. , 1936 , 42 , 275–283.
doi:10.1090/S0002-9904-1936-06290-7

  8. Hamilton , H. J. Transformations of multiple sequences. Duke Math. J. , 1936 , 2 , 29–60.
doi:10.1215/S0012-7094-36-00204-1

  9. Pringsheim , A. Elementare Theorie der unendliche Doppel-reihen. Sitsungsber. Akad. Wiss. München , 1897 , 27 , 101–153.

10. Sheffer , I. M. Note on multiply-infinite series. Bull. Amer. Math. Soc. , 1946 , 52 , 1036–1041.
doi:10.1090/S0002-9904-1946-08706-6

11. Vorob¢ev , N. N. Theory of Series , 5th edn. Nauka , Moscow , 1986 (in Russian).

12. Zeltser , M. Investigation of Double Sequence Spaces by Soft and Hard Analytical Methods. PhD~thesis , Tartu Univ. Press , Tartu , 2001.

 
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Current Issue: Vol. 68, Issue 3 in Press, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December