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A Distribution Function of Cantor-Vitali Type

Published online by Cambridge University Press:  20 November 2018

D. B. Sumner
D. B. Sumner*
Affiliation:
McMaster University
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In his 1922 article [l] on functions of bounded variation, Vitali gave a method for constructing monotone non-absolutely continuous functions, generalizing ideas from the ternary set introduced in another connection by Cantor. In [2], Hille and Tamarkin gave a full account of the "middle-third" function, showing it to be a singular distribution function, and finding its characteristic function. In [3], Evans obtained a generalization of the middle - third function by discarding middle intervals of length other than one-third, and obtained algorithms by which the moments of his function could be calculated. Invarious papers, among them [4], Wintner studied infinite convolutions of symmetric Bernoulli distributions, finding a great variety of distributions whose characteristic functions were of the form

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 1 , March 1964 , pp. 65 - 75
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Vitali, G., Analisi delle funzioni a varia zione limitata. Rend, del Cire. Mat. di Palermo, Vol. 46 (1922), 388–408.Google Scholar
2. Hille, E. and Tamarkin, J. D., Remarks on a known example of a monotone continuous function. Amer. Math. Monthly, Vol. XXXVI (1929), No. 5, 255–264.Google Scholar
3. Evans, G. C., Calculation of moments for a Cantor-Vitali function, Herbert Ellsworth Slought Memorial Paper No. 6. Supp. Amer. Math. Monthly 64 (1957), No. 8, 22–27.Google Scholar
4. Jessen, B. and Wintner, A., Distribution functions and the Riemann Zeta function. Trans. Amer. Math. Soc., Vol. 38 (1935), 48–88.Google Scholar