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Examples for the Theory of Infinite Iteration of Summability Methods

Published online by Cambridge University Press:  20 November 2018

Persi Diaconis*
Affiliation:
Stanford University, Stanford, California
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Abstract

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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.

Type
Research Article
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
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6. Flehinger, B., On the probability a random integer has initial digit A, Amer. Math. Month. 73 (1966), 10561061.Google Scholar
7. Garten, V. and Knopp, K., Ungleichungen Zwischen Mittlewerten von Zahlenfolgen und Funktionen, Math. Z. 42 (1937), 365388.Google Scholar
8. Golomb, S., A class of probability distributions on the integers, J. Number Theory 2 (1970), 189192.Google Scholar
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12. Serre, J. P., A course in arithmetic (Springer, New York, 1973).CrossRefGoogle Scholar
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