THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENC 



[FIFTH SERIES.] 



AUGUST 1896. 



XI. On the Convergence of Fourier s Series. By W. Williams, 

 B.Sc, Royal College of Science, London* . 



I. 



rpHE convergency of Fourier's series is a subject which . 

 *- has been so fully investigated by eminent mathemati- 

 cians that it is necessary to offer some apology for venturing 

 to discuss it afresh. It is, however, a subject of such singular 

 difficulty, — a difficulty which has only been partially over- 

 come, — and the investigations connected with it are so 

 laborious and abstruse in character, that any simplification 

 that may be effected in the method of attack is of value quite 

 apart from any fresh light that may be thrown upon the con- 

 vergency itself. The chief difficulty connected with the 

 investigation is that of assigning the necessary conditions to 

 be fulfilled by the function which determines the coefficients 

 of the series, and this difficulty arises from the highly general 

 manner in which the term " function " is defined and employed 

 in modern analysis. Of course, if we confine ourselves to 

 the comparatively simple functions which occur in the prac- 

 tical applications of the series, functions, for example, which 

 are continuous and obey the laws of the differential calculus, 

 much of this difficulty disappears. But it is necessary that 

 we should, in such a case, state clearly the limitation which 

 we make, as otherwise our investigation partakes of a too 

 general character, and proves too much. For, as we shall 



* Communicated by the Author. 

 Phil. Mag. S. 5. Vol. 42. No. 255. Aug. 1896. L 



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