|ISSN 1088-6826(online) ISSN 0002-9939(print)|
Summability methods fail for the partial sums of Fourier series
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 th partial sums of the Fourier series, there is no summability method whatever which produces convergence for all continuous functions.
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