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Summability methods fail for the $2^{n}th$ partial sums of Fourier series

Author: D. J. Newman
Journal: Proc. Amer. Math. Soc. 45 (1974), 300-302
MSC: Primary 42A24
MathSciNet review: 0358200
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Abstract: Although the Fourier series of a continuous function need not converge everywhere, it was an important discovery of Fejér that this series must be Cesàro summable. Indeed, it is a frequent occurrence that convergence may be restored to an expansion by use of an appropriate summability method. What we show in this note is that the very opposite phenomenon can occur. Namely, that if one considers only the ${2^n}$th partial sums of the Fourier series, there is no summability method whatever which produces convergence for all continuous functions.

References [Enhancements On Off] (What's this?)

  • R. E. Edwards, Fourier series: a modern introduction. Vol. II, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1967. MR 0222538

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Keywords: Summability, Fourier series
Article copyright: © Copyright 1974 American Mathematical Society