Abstract
Various conditions have been claimed in a recent paper as necessary for the convergence of a Fourier series at a point, principally the Lebesgue summability of the series. These conditions are, however, shown to be necessary only in a particular class of functions and the error in the argument is explicated. The relevant facts about Lebesgue summability are reviewed.
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References
N. K. Bary, A treatise on trigonometric series, Macmillan ( New York, 1964).
P. Chandra, Necessary conditions for the convergence of Lebesgue-Fourier series, Analysis Math., 24(1998), 171–180.
G. H. Hardy and J. E. Littlewood, Notes on the theory of series. VII: On Young's convergence criterion for Fourier series, Proc. London Math. Soc., 28(1928), 301–311.
S. Izumi, N. Matsuyama, and T. Tsuchikara, Some negative examples, Tôhoku Math. J., 5(1953), 43–51.
D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math., 44(1972), 107–117.
D. Waterman, On Λ-bounded variation, Studia Math., 57(1976), 33–45.
D. Waterman, Fourier series of functions of ?-bounded variation, Proc. Amer. Math. Soc., 74(1979), 119–123.
D. Waterman, A generalization of the Salem test, Proc. Amer.Math. Soc., 105(1989), 129–133.
A. Zygmund, Trigonometric Series, University Press ( Cambridge, 1959).
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Waterman, D. Necessary Conditions for the Convergence of Fourier Series. Analysis Mathematica 26, 235–239 (2000). https://doi.org/10.1023/A:1005609319903
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DOI: https://doi.org/10.1023/A:1005609319903