Abstract
1. The necessary and sufficient condition that a trigonometrical series should be a Fourier series is that the integrated series should converge to an integral throughout the closed interval of periodicity, and should be the Courier series, accordingly, of an integral. Conversely, starting with the Courier series of an integral and differentiating it term by term, we obtain the Courier series of the most general type, namely, one associated with any function possessing an absolutely convergent integral. If the Courier series which is differentiated is not the Courier series of an integral, but of a function which fails to be an integral, at even a single point, the derived series will not lie a Courier series.
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Schedule a callMore From: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character
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