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On Hausdorff's methods of summability

Published online by Cambridge University Press:  24 October 2008

W. W. Rogosinski
W. W. Rogosinski
Affiliation:
Aberdeen, King's College August 1941
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Extract

The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transforms

of a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 38 , Issue 2 , April 1942 , pp. 166 - 192
Copyright
Copyright © Cambridge Philosophical Society 1942

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References

REFERENCES

(1)Carlson, F. Sur une classe de séries de Taylor, Thesis (Upsala, 1914).Google Scholar
(2)Hausdorff, F.Summationsmethoden und Momentfolgen (I). Math. Z. 9 (1921), 74–109.CrossRefGoogle Scholar
(3)Hurwitz, W. A. and Silverman, L. L.On the consistency and equivalence of certain definitions of summability. Trans. American Math. Soc. 18 (1917), 1–20.CrossRefGoogle Scholar
(4)Knopp, K. Grenzwerte von Reihen bei Annäherung an die Konvergenzgrenze, Dissertation (Berlin, 1907).Google Scholar
(5)Mercer, J.On the limits of real variants. Proc. London Math. Soc. (2), 5 (1907), 206–24.CrossRefGoogle Scholar
(6)Paley, R. E. A. C. and Wiener, N.Fourier transforms in the complex domain, Amer. Math. Soc. Colloquium Publ., vol. 19 (New York, 1934).Google Scholar
(7)Pitt, H. R.Mercerian theorems. Proc. Cambridge Phil. Soc. 34 (1938), 510–20.CrossRefGoogle Scholar
(8)Pitt, H. R.General Tauberian theorems. Proc. London Math. Soc. 44 (1938), 243–88.CrossRefGoogle Scholar
(9)Schnee, W.Die Identität des Cesàroschen und Hölderschen Grenzwertes. Math. Ann. 67 (1909), 110–25.CrossRefGoogle Scholar
(10)Silverman, L. L. and Tamarkin, J. D.On the generalisation of Abel's Theorem for certain definitions of summability. Math. Z. 29 (1929), 161–170.CrossRefGoogle Scholar
(11)Titchmarsh, E. C.The theory of functions (second edition, Oxford, 1939).Google Scholar
(12)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, 1937).Google Scholar
(13)Toeplitz, O.Über allgemeine lineare Mittelbildungen. Prace Matematyczno-Fisyczne, 22 (1911), 113–19.Google Scholar
(14)Wiener, N.A new method in Tauberian theorems. Massachusetts J. Math. Phys. 7 (1928), 161–84.CrossRefGoogle Scholar
(15)Wiener, N.Tauberian Theorems. Ann. Math. 33 (1932), 1–100.CrossRefGoogle Scholar
(16)Wiener, N. and Pitt, H. R.On absolutely convergent Fourier-Stieltjes transforms. Duke Math. J. 4 (1938), 420–36.CrossRefGoogle Scholar