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The Distribution of Sequences and Summability

Published online by Cambridge University Press:  20 November 2018

A. F. Dowidar  and
G. M. Petersen
A. F. Dowidar
Affiliation:
University College of Swansea
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We suppose that 0 < sn ≤ 1 for every n, and denote by n(α, β) the number of S0, S1, S2, . . . , Sn which fall in the interval 0 ≤ α < x ≤ β ≤ 1. If there exists a function g(t), 0 ≤ t ≤ 1, such that

for every interval (α, β] with 0 ≤ β — α ≤ 1, the sequence (Sn) is said to have a distribution function g(t), 0 ≤ t ≤ 1, in the interval [0, 1], (see 9, p. 87).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 1 - 10
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Cooke, R. G. and Barnett, A. M., The “right” value for the generalized limit of a bounded divergent sequence, J. London Math. Soc, 23 (1948), 211–221.Google Scholar
2. Cooke, R. G., Infinite matrices and sequence spaces (London, 1950).Google Scholar
3. Halmos, P. R., Measure theory (Princeton, 1950).Google Scholar
4. Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
5. Henstock, R., The efficiency of matrices for bounded sequences, J. London Math. Soc, 25 (1950), 27–33.Google Scholar
6. Keogh, F. R., Lawton, B., and Petersen, G. M., Well distributed sequences, Can. J. Math. 10 (1958), 572–576.Google Scholar
7. Knopp, K., Mengentheoretische Behandlung einiger Problème der diophantischen Approximationen und der transfiniten Wahrscheinlichkeiten, Math. Ann., 95 (1926), 409–426.Google Scholar
8. Koksma, J. F., Diophantische Approximationen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. IV (Berlin, 1936).Google Scholar
9. Lorentz, G. G., Borel and Banach properties of methods of summation, Duke Math. J. 22 (1955), 129–141.Google Scholar
10. Natanson, I. P., Theory of functions of a real variable, Translated by Boron and Hewitt from the Russian (New York, 1955).Google Scholar
11. Niven, I., Irrational numbers, Carus Monographs, no. 11 (1956).Google Scholar
12. Petersen, G. M., Almost convergence and uniformly distributed sequences, Quart. J. Math. (Oxford), 7 (1956), 188–191.Google Scholar
13. Wall, H. S., Analytic theory of continued fractions (New York, 1948).Google Scholar
14. Weyl, H., Ueber die Gleichverteilung von Zahlen mod. Kins, Math. Ann., 77 (1916), 313–352.Google Scholar
15. Zygmund, A., Trigonometric series, Vol. I (Cambridge, 1959), see p. 140.Google Scholar