184 nR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



it is thus clear thftt the canonical Sturm-Liouville series will all converge uniformly 

 to/(s) in (y, 8), when 



converges uniformly to zero in this interval. We have thus proved the theorem : 



The answer to the question whether the canonical Sturm- Lioumlle series corre- 

 tpondinf/ to an assigned function all converge uniformly, or not, in an interval (y, 8) 

 contained urithin (0, IT) depends only upon the values assumed by the function in an 

 interval enclosing (y, 8) in it* interior, and exceeding/ it in lent/th by an arbitrarily 

 small amount. 



It is evidently a necessary condition for uniform convergence that the function 

 should be continuous in (y, 8). 



18. After what was said in the preceding paragraph, it will be clear that, 



when lim s, (a) exists as a finite number at a point of the open interval (0, IT), all 



-* 



the canonical Stunn-Liouville series corresponding to /(.<?) converge at this point, and 

 have the value of this limit for their common sum. We may employ this fact to 

 obtain conditions under which these series converge. 



Let <>(*), the bilateral limit of f(s) at the point s, exist as a finite number. In 

 virtue of (35), it is easily seen that 



t. ()- () gg - [/. (s-t) + f t (,-H)-2 ()] Sn - dt. . (36) 



TT . SHI "5"* 



Let us now suppose that, as ft diminishes indefinitely, the integral 



sin 



converges uniformly to zero for positive integral values of n. When any positive 

 number e is assigned, we may then choose a in such a way that the numerical value 

 of the right-hand member of (36) is less than e for these values of n. Further, a 

 Ijeing fixed, a positive integer N may be chosen great enough to ensure that the 

 difference between the left- and right-hand members is numerically less than e, for 

 n i N. Thus we have 



5. () -(*)! 



for a N ; and therefore 



lim S B (s) = w 



We have now established the theorem : 



At a point s of the open, interval (0, ), where (,), the bilateral limit of f(*) t 

 te numl>er, a sufficient condition that all the canonical Sturm-Liourille 



