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Theorems concerning the summability of series by boreľs exponential method

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Rendiconti del Circolo Matematico di Palermo (1884-1940)

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References

  1. G. H. Hardy andJ. E. Littlewood,The Relations between Borel’s and Cesàro’sMethods of Summation [Proceedings of the London Mathematical Society, series II, vol. XI (1912-1913), pp. 1–16].

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  2. For an explanation of our reasons for giving this name to the theorem, seeG. H. Hardy andJ. E. Littlewood,Contributions to the arithmetic Theory of Series [Proceedings of the London Mathematical Society, series II, vol. XI (1912–1913), pp. 411–478 (p. 413)].

  3. SeeG. H. Hardy andM. Riesz,The general Theory of Dirichlet’s Series [Cambridge Mathematical Tracts, no. 18, 1915], p. 56.

  4. We cannot quote any general theorem of which this equation is a direct corollary: but the materials necessary for the proof will be found in our paper « Contributions, etc. », loc. cit.2), pp. 452 et seq.

  5. G. H. Hardy,The Application to Dirichlet’sSeries of Borel’sexponential Method of Summation [Proceedings of the London Mathematical Society, series 11, vol. VIII (1910), pp. 277–294].

  6. G. H. Hardy,Researches in the Theory of divergent Series and divergent Integrals [Quarterly Journal, vol. XXXV (1904), pp. 22–66], p. 40; T. J. ľA. Bromwich,Infinite Series, pp. 319–322.

  7. M. Riesz,Sur la représentation analytique des fonctions définies par des séries de Dirichlet [Acta Mathematica, t. XXXV (1912), pp. 253–270].

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    G. H. Hardy & J. E. Littlewood

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  1. G. H. Hardy
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  2. J. E. Littlewood
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Hardy, G.H., Littlewood, J.E. Theorems concerning the summability of series by boreľs exponential method. Rend. Circ. Matem. Palermo 41, 36–53 (1916). https://doi.org/10.1007/BF03018286

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