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# On the Rate of Growth of the Walsh Antidifferentiation Operator

R. Penney
Proceedings of the American Mathematical Society
Vol. 55, No. 1 (Feb., 1976), pp. 57-61
DOI: 10.2307/2041841
Stable URL: http://www.jstor.org/stable/2041841
Page Count: 5
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 $x \rightarrow 0$. We use this information to study continuity properties of differentiable functions.