Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the rate of growth of the Walsh antidifferentiation operator


Author: R. Penney
Journal: Proc. Amer. Math. Soc. 55 (1976), 57-61
DOI: https://doi.org/10.1090/S0002-9939-1976-0397289-0
MathSciNet review: 0397289
Full-text PDF Free Access

Abstract | References | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Butzer and Wagner, Walsh series and the concept of a derivative, Applicable Anal. 3 (1973), 29-46. MR 0404978 (53:8774)
  • [2] -, On a Gibbs-type derivative in Walsh-Fourier analysis with applications, Technical Report of the Technological University of Aachen, Aachen, West Germany.
  • [3] Powell and Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972.
  • [4] S. Yano, On Walsh-Fourier series, Tôhoko Math. J. (2) 3 (1951), 223-242. MR 13, 550. MR 0045236 (13:550a)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0397289-0
Keywords: Walsh functions, antidifferentiation
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society