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

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

MathSciNet review:
1123663

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Abstract: We consider the Walsh-Fourier series of a function assumed to be of bounded fluctuation on the interval . 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 at a point 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.

**[1]**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****[2]**C. W. Onneweer,*On uniform convergence for Walsh-Fourier series*, Pacific J. Math.**34**(1970), 117–122. MR**0275048****[3]**C. W. Onneweer and Daniel Waterman,*Fourier series of functions of harmonic bounded fluctuation on groups*, J. Analyse Math.**27**(1974), 79–83. MR**0481938**, https://doi.org/10.1007/BF02788643**[4]**R. E. A. C. Paley,*A remarkable system of orthogonal functions*, Proc. London Math. Soc.**34**(1932), 241-279.**[5]**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****[6]**J. L. Walsh,*A Closed Set of Normal Orthogonal Functions*, Amer. J. Math.**45**(1923), no. 1, 5–24. MR**1506485**, https://doi.org/10.2307/2387224**[7]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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Additional Information

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

Keywords:
Rademacher functions,
Walsh functions in the Paley enumeration,
Walsh-Fourier series,
pointwise convergence,
rate of convergence,
uniform convergence,
-continuity,
bounded fluctuation,
bounded variation,
Walsh-Dirichlet kernel,
Dirichlet-Jordan test

Article copyright:
© Copyright 1993
American Mathematical Society