On the rate of growth of the Walsh antidifferentiation operator
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- by R. Penney PDF
- Proc. Amer. Math. Soc. 55 (1976), 57-61 Request permission
Abstract:
In [1] Butzer and Wagner introduced a concept of differentiation and antidifferentiation of Walsh-Fourier series. Antidifferentiation is accomplished by convolving (in the sense of the Walsh group) against a function $\Omega$. In this paper we study growth and the continuity properties of $\Omega$ showing that $\Omega$ is bounded from below by $- 1$, is continuous in $(0,1)$ and grows at most like $\log 1/x$ as $x \to 0$. We use this information to study continuity properties of differentiable functions.References
- P. L. Butzer and H. J. Wagner, Walsh-Fourier series and the concept of a derivative, Applicable Anal. 3 (1973), 29–46. MR 404978, DOI 10.1080/00036817308839055 —, On a Gibbs-type derivative in Walsh-Fourier analysis with applications, Technical Report of the Technological University of Aachen, Aachen, West Germany. Powell and Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972.
- Shigeki Yano, On Walsh-Fourier series, Tohoku Math. J. (2) 3 (1951), 223–242. MR 45236, DOI 10.2748/tmj/1178245527
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 57-61
- DOI: https://doi.org/10.1090/S0002-9939-1976-0397289-0
- MathSciNet review: 0397289