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A differentiation theorem for functions defined on the dyadic rationals

Author: R. J. Lindahl
Journal: Proc. Amer. Math. Soc. 30 (1971), 349-352
MSC: Primary 26.40; Secondary 42.00
MathSciNet review: 0284549
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Abstract: In this paper we show that under certain conditions a real-valued function defined on an interval of dyadic rational numbers is a monotone function. One of these conditions involves a generalized differentiability property. From this result we offer a new proof of a conjecture of N. Fine concerning the uniqueness of solution of Walsh series.

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  • [1] R. B. Crittenden and V. L. Shapiro, Sets of uniqueness on the group $ {2^\omega }$, Ann. of Math. (2) 81 (1965), 550-564. MR 31 #3783. MR 0179535 (31:3783)
  • [2] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. MR 11, 352. MR 0032833 (11:352b)
  • [3] S. Saks, Théorie de l'intégrale, Monografie Mat., vol. 2, PWN, Warsaw, 1933; English transl., Monografie Mat., vol. 7, PWN, Warsaw; Hafner, New York, 1937.

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Keywords: Monotone functions on dyadic rationals, Walsh series, Walsh-Fourier series
Article copyright: © Copyright 1971 American Mathematical Society

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