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Certain Groups of Orthonormal Step Functions

Published online by Cambridge University Press:  20 November 2018

J. J. Price
J. J. Price*
Affiliation:
University of Pennsylvania
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It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 413 - 425
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Chrestenson, H. E., A class of generalized Walsh functions, Pacific J. Math., 5 (1955), 17–31.Google Scholar
2. Fine, N. J., On the Walsh functions, Trans. Amer. Math. Soc, 65 (1949), 372–414.Google Scholar
3. Ohkuma, T., On a certain system of orthogonal step functions, Tohoku Math. J., 5 (1953), 166–177.Google Scholar
4. Vilenkin, N., On a class of complete orthonormal systems, Bull. Acad. Sci. URSS. Ser. Math. (Izvestia Akad. Nauk. SSSR), 11 (1947), 363–400 (Russian, English summary).Google Scholar
5. Walsh, J. L., A closed set of normal orthogonal functions, Amer. J. Math., 55 (1923), 5–24.Google Scholar