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

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.

References

R. Bojanić, An estimate of the rate of convergence for Fourier series of functions of bounded variation, Publ. Inst. Math. (Beograd) (N.S.) 26(40) (1979), 57–60. MR 572330
C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117–122. MR 275048, DOI 10.2140/pjm.1970.34.117
C. W. Onneweer and Daniel Waterman, Fourier series of functions of harmonic bounded fluctuation on groups, J. Analyse Math. 27 (1974), 79–83. MR 481938, DOI 10.1007/BF02788643

A remarkable system of orthogonal functions

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F. Schipp, W. R. Wade, and P. Simon, Walsh series, Adam Hilger, Ltd., Bristol, 1990. An introduction to dyadic harmonic analysis; With the collaboration of J. Pál. MR 1117682
J. L. Walsh, A Closed Set of Normal Orthogonal Functions, Amer. J. Math. 45 (1923), no. 1, 5–24. MR 1506485, DOI 10.2307/2387224
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776