A quantitative Dirichlet-Jordan test for Walsh-Fourier series

Móricz, Ferenc

doi:10.1090/S0002-9939-1993-1123663-0

A quantitative Dirichlet-Jordan test for Walsh-Fourier series
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by Ferenc Móricz

Proc. Amer. Math. Soc. 118 (1993), 143-149

DOI: https://doi.org/10.1090/S0002-9939-1993-1123663-0

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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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