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On convergence and summability factors in a sequence

Published online by Cambridge University Press:  26 February 2010

L. S. Bosanquet
L. S. Bosanquet
Affiliation:
University College, London.
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Extract

The following theorem is typical of a group of results which have been found to be of importance in the theory of Cesàro summability.

Theorem A. If 0 ≤ ρ ≤ κ (κ, ρ integer), p > −1,p+q −1 and

and ifsn = 0(np) (C, κ), thensn εn = 0(np+q) (C, ρ).

Type
Research Article
Information
Mathematika , Volume 1 , Issue 1 , June 1954 , pp. 24 - 44
Copyright
Copyright © University College London 1954

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References

1Anderson, A. F., Studier over Cesà's summabilitetsmetode (Copenhagen, 1921).Google Scholar
2Anderson, A. F. “Comparison theorems in the theory of Cesaro summability”, Proc. London Math. Soc. (2), 27 (1928), 39–71.CrossRefGoogle Scholar
3Bohr, H., “Sur la série de Dirichlet”, Comptes Rendus, 148 (1909), 75–80; reprinted in (5, Vol. I).Google Scholar
4Bohr, H., Bidrag til de Dirichlet'ske Rcekker's Theori (Copenhagen, 1910); reprinted in (5, Vol. I); English translation in (5, Vol. III).Google Scholar
5Bohr, H., Collected Mathematical Works, edited by , E. Fφlner and Jessen, B. (Copenhagen, 1952).Google Scholar
6Borwein, D., “On the Cesàro summability of integrals”, Journal London Math. Soc., 25 (1950), 289–302.CrossRefGoogle Scholar
7Bromwich, T. J. I'A., “On the limits of certain infinite series and integrals”, Math. Annalen, 65 (1908), 350–369.CrossRefGoogle Scholar
8Bosanquet, L. S., “Note on differentiated Fourier series”, Quart. J. of Math. (Oxford, 1), 10 (1939), 67–74.CrossRefGoogle Scholar
9Bosanquet, L. S., “The absolute summability problem for differentiated Fourier series”, Quart. J. of Math. (Oxford, 1), 12 (1941), 15–25.CrossRefGoogle Scholar
10Bosanquet, L. S., “Note on the Bohr–Hardy theorem”, Journal London Math. Soc., 17 (1942), 166–173.CrossRefGoogle Scholar
11Bosanquet, L. S., “Some properties of Cesaro–Lebesgue integrals”, Proc. London Math. Soc. (2), 49 (1945–9), 40–62.Google Scholar
12-14Bosanquet, L. S., “Note on convergence and summability factors (I–III)”, Journal London Math. Soc., 20 (1945), 39–48;CrossRefGoogle Scholar
Proc. London Math. Soc. (2), 50 (1948–9), 295–304, and Proc. London Math. Soc. 482–496.Google Scholar
15Bosanquet, L. S., “On convergence and summability factors in a Dirichlot series (II)”, Journal London Math. Soc., 23 (1948), 35–38.CrossRefGoogle Scholar
16Bosanquet, L. S., “Note on a theorem of M. Riesz”, Proc. London Math. Soc.. (3), 1 (1951), 453–461.CrossRefGoogle Scholar
17Bosanquet, L. S., and Chow, H. C., “Some analogues of a theorem of Andersen”, Journal London Math. Soc., 16 (1941), 42–48.CrossRefGoogle Scholar
18Bosanquet, L. S., and Hyslop, J. M., “On the absolute summability of the allied series of a Fourier series”, Math. Zeitschrift, 42 (1937), 489–512.CrossRefGoogle Scholar
19Fekete, M., “Summabilitàsi factor–sorozatok”, Math, es Termés. Ert., 35 (1917), 309–324.Google Scholar
20Hadamard, J., “Deux théorèmes d'Abel sur la convergence des séries”, Acta Math., 27 (1903), 177–183.CrossRefGoogle Scholar
21Hardy, G. H., “Generalisation of a theorem in the theory of divergent series”, Proc. London Math. Soc. (2), 6 (1908), 255–264, and Acta Math., 8 (1910), footnote, 279–281.CrossRefGoogle Scholar
22Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
23Hardy, G. H., and Littlewood, J. E., “A theorem in the theory of a summable divergent series”, Proc. London Math. Soc. (2), 27 (1928), 327–348.CrossRefGoogle Scholar
24Kojima, T., “On the generalized Toeplitz's theorems on limit and their applications”, Tôhoku Math. J., 12 (1917), 291–326.Google Scholar
25Schur, J., “Uber lineare Transformationen in der Theorie der unendlichen Reihen”, Journal fàr Math., 151 (1921), 79–111.Google Scholar