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Examples for the Theory of Infinite Iteration of Summability Methods

  • Persi Diaconis (a1)
Abstract

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 ﹛aii=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.

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References
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1. Diaconis, P., Weak and strong averages in probability and the theory of numbers, Ph.D. Dissertation, Department of Statistics, Harvard University, 1974.
2. Duran, J. P., Almost convergence, summability and ergodicity, Can. J. Math. 26 (1974), 372387.
3. Eberlein, W. F., Banach-Hausdorff limits, Proc. Amer. Math. Soc. 1 (1950), 662665.
4. Eberlein, W. F. On Holder summability of infinite order, Notices Amer. Math. Soc. 19 (1972), A-164; Abstract No. 691–46-21.
5. Feller, W., An introduction to probability theory and its applications, Vol. II, second ed. (Wiley, New York, 1971).
6. Flehinger, B., On the probability a random integer has initial digit A, Amer. Math. Month. 73 (1966), 10561061.
7. Garten, V. and Knopp, K., Ungleichungen Zwischen Mittlewerten von Zahlenfolgen und Funktionen, Math. Z. 42 (1937), 365388.
8. Golomb, S., A class of probability distributions on the integers, J. Number Theory 2 (1970), 189192.
9. Hardy, G. H., Divergent series (Oxford University Press, Oxford, 1949).
10. Hasse, H., Vorlesungen Uber Zahlentheorie (Springer, Berlin, 1950).
11. Ishiguro, K., Tauberian theorems concerning summability methods of log type, Proc. Japan. Acad. 39 (1963), 156159.
12. Serre, J. P., A course in arithmetic (Springer, New York, 1973).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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