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A quantitative Dirichlet-Jordan test for Walsh-Fourier series

Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 118 (1993), 143-149
MSC: Primary 42C10; Secondary 41A30
MathSciNet review: 1123663
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Abstract: We consider the Walsh-Fourier series $ \sum {{a_k}{w_k}(x)} $ of a function $ f$ assumed to be of bounded fluctuation on the interval $ [0,1)$. Every function of bounded variation is also of bounded fluctuation on the same interval, but not conversely. We present an estimate for the difference of $ f(x)$ at a point $ x \in [0,1)$ and the partial sum of its Walsh-Fourier series in terms of the bounded fluctuation operator. This gives rise to a local convergence result. As special cases, we obtain a Walsh analogue of the Dirichlet-Jordan test and a global convergence result due to Onneweer.

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Keywords: Rademacher functions, Walsh functions in the Paley enumeration, Walsh-Fourier series, pointwise convergence, rate of convergence, uniform convergence, $ W$-continuity, bounded fluctuation, bounded variation, Walsh-Dirichlet kernel, Dirichlet-Jordan test
Article copyright: © Copyright 1993 American Mathematical Society