Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T05:54:20.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples for the Theory of Infinite Iteration of Summability Methods

Published online by Cambridge University Press:  20 November 2018

Persi Diaconis
Persi Diaconis*
Affiliation:
Stanford University, Stanford, California
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Garten and Knopp [7] introduced the notion of infinite iteration of Césaro (C1 ) averages, which they called H∞ summability. Flehinger [6] (apparently unaware of [7]) produced the first nontrivial example of an H∞ summable sequence: the sequence ﹛ai ﹜ ∞i=1 where at is 1 or 0 as the lead digit of the integer i is one or not. Duran [2] has provided an elegant treatment of H∞ summability as a special case of summability with respect to an ergodic semigroup of transformations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 3 , 01 June 1977 , pp. 489 - 497
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Diaconis, P., Weak and strong averages in probability and the theory of numbers, Ph.D. Dissertation, Department of Statistics, Harvard University, 1974.Google Scholar
2. Duran, J. P., Almost convergence, summability and ergodicity, Can. J. Math. 26 (1974), 372–387.Google Scholar
3. Eberlein, W. F., Banach-Hausdorff limits, Proc. Amer. Math. Soc. 1 (1950), 662–665.Google Scholar
4. Eberlein, W. F. On Holder summability of infinite order, Notices Amer. Math. Soc. 19 (1972), A-164; Abstract No. 691–46-21.Google Scholar
5. Feller, W., An introduction to probability theory and its applications, Vol. II, second ed. (Wiley, New York, 1971).Google Scholar
6. Flehinger, B., On the probability a random integer has initial digit A, Amer. Math. Month. 73 (1966), 1056–1061.Google Scholar
7. Garten, V. and Knopp, K., Ungleichungen Zwischen Mittlewerten von Zahlenfolgen und Funktionen, Math. Z. 42 (1937), 365–388.Google Scholar
8. Golomb, S., A class of probability distributions on the integers, J. Number Theory 2 (1970), 189–192.Google Scholar
9. Hardy, G. H., Divergent series (Oxford University Press, Oxford, 1949).Google Scholar
10. Hasse, H., Vorlesungen Uber Zahlentheorie (Springer, Berlin, 1950).Google Scholar
11. Ishiguro, K., Tauberian theorems concerning summability methods of log type, Proc. Japan. Acad. 39 (1963), 156–159.Google Scholar
12. Serre, J. P., A course in arithmetic (Springer, New York, 1973).Google Scholar