122 



SCIENCE 



[N. S. Vol. XXXIX. No. 995 



applications has repeatedly served as a 

 guide post. Numberless analogous results 

 have been suggested thereby, though with- 

 out definite statement of the fact. To 

 take an example at random, the relation 



f' Pi,{x)PiUx)dx = (m + n) 



Jo 



has its trigonometric analogues 



r 



Jo 



COS n$ cos mS dO = 0. 



Who can deny, or who can affirm, in many 

 such individual instances that the sugges- 

 tion came from the trigonometric series? 

 Yet in the bulk the debt is so great that he 

 who runs can read it. 



It is especially in connection with bound- 

 ary value problems that we encounter 

 series of functions. Now the trigonometric 

 series was the inevitable tool for the first 

 boundary value problems — those of vibrat- 

 ing strings, rods, columns of air, etc. Later, 

 when Fourier crystallized the boundary . 

 value problems into classic shape, he used 

 trigonometric series and, to lesser degree, 

 similar series of Bessel's functions, obvi- 

 ously because these afforded him the sim- 

 plest tools for the simplest problems. From 

 series of sines of multiple angles he was 

 led by certain problems in heat conduction 

 to series of form Scjsin aiX, where the 

 tti are roots of a certain transcendental 

 equation. Thence the orientizing influence 

 of Fourier's series is continued down to 

 the modern development of normal func- 

 tions in the theory of integral equations. 

 All such influences are in the very warp 

 and woof of mathematical development and 

 can not be disentangled. To minimize or 

 ignore them would be to give a distorted 

 picture. They form a most vital and lead- 

 ing part of the mighty theory of harmonic 

 and normal functions and of the boundary 

 value theory. 



The extent of these influences in the past 

 gives rise naturally to the question of 



whether the trigonometric series will con- 

 tinue to exert such a moulding influence in 

 the future. Certain results of Baire to be 

 shortly mentioned incline one to answer 

 negatively. Yet the questions regarding the 

 convergence of the series and the character 

 of the functions which it can represent are 

 even to-day incompletely answered. When 

 new implements are invented, it is still to 

 these unanswered questions that the investi- 

 gator naturally turns to test their worth, as, 

 for example, Lebesgue with his great new 

 concept of an integral which has applica- 

 tion when Riemann's integral is void of 

 sense, or Fejer with a method of summing a 

 divergent series. Also the Fourier series 

 still offers an occasional surprise. Who in- 

 deed would have anticipated Gibbs's dis- 

 covery, since extended by Bocher, which re- 

 lates to the approximation curve y=^S„(x), 

 obtained by equating y to the sum of the 

 first n terms of the series (2) above? As 

 n increases indefinitely, the amount of the 

 oscillation of the curve in the vicinity of 

 each point of discontinuity of the limit does 

 not tend toward the measure of the discon- 

 tinuity, as would be supposed, but to this 

 value increased in a certain definite ratio ! 

 But it may be reasonably expected that 

 these surprises will become fewer and less 

 important. 



In this brief review I have neglected 

 certain less analytic aspects, such as trig- 

 onometric interpolation and the use of the 

 series in computation and in the pertur- 

 bation theory. It has also not been neces- 

 sary to emphasize the simplicity of structure 

 of the series and its adaptation to compu- 

 tation. Neither do I need to speak of its 

 correspondence in structure to so many 

 periodic phenomena of nature, sound, light, 

 the tides, etc. But I do wish, in closing, 

 to emphasize and examine further, one 

 aspect implied in all my preceding con- 



