418 REPORTS ON THE STATE OP SCIENCE. 



& + *■** 



and the interval extends to inlinity. The characteristic values of X are 

 then everywhere dense along the positive half of the X axis, and so 

 form a continuous spectrum or a band spectrum. Fourier's integral 

 formula now takes the place of the expansion theorem. The representa- 

 tion of arbitrary functions by means of the normal functions of a 

 differential equation with a singular point at the end of the interval is 

 very complex. In addition to a continuous spectrum a point spectrum 

 can also occur : we thus have a mixed representation, partly by a series 

 formed according to Fourier's rule and partly by an integral analogous 

 to Fourier's double integral. It can also happen that an integral 

 representation is obtained in which the integral is taken over several 

 separate intervals or bands, and the isolated points of the spectrum are 

 in the different intervals between the bands. 



Wirtinger l first came across such a distribution of singular values 

 in the case of a vibrating string of infinite length ; he called the 

 distribution a band spectrum. The discovery of the existence of a band 

 spectrum in the year 1897 is quite noteworthy. 



The general theory of band spectra has been given by Hilbert 2 in 

 his theory of quadratic forms in an infinite number of variables. This 

 theory seems to cover all the cases that have so far been treated, and it 

 must be considered a very noteworthy achievement on Hilbert's part to 

 have established theorems of such very wide application. 



The differential equations to which the theory has been applied are 

 of the form 



L(«) +\u = j- \s%s) d ~~] - q(s)u + Xm = 



where l(s), q(s) behave regularly as analytic functions in the neighbour- 

 hood of the strip ofEsfEl of the real axis, and l(s) is everywhere positive 

 for o5s5l; also l(o) = 1. At the end point s = 1 an arbitrary homo- 

 geneous boundary condition is ascribed. 



The theory is due chiefly to Hilb ;i and Weyl, 4 who obtain a repre- 

 sentation of the form 



a 



CO CO 



+ 



the integrals being absolutely and uniformly convergent if 

 i i i 



[f(s)Ych, \[L(f)] 2 ds |[ 



ds 



converge. The functions \p t (s) i£(s,X) are solutions of the differential 

 equation for appropriate values of X. Hilb's results have also been 



1 Math. Ann., 1807, Bd. 48, p. 387. 



-' Gottingm Nachrichten, 190G, pp. 157-227, pp. 439-80. 



3 Math, Ann., 1908, Bd. 66, p. 1. 4 Ibid., 1910, Bd. 68, p. 220. 



