IT NOTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 



191 



Again, let us suppose that the set of points at which f(s) is continuous includes a 

 closed interval (y, 8) lying within (0, TT). Then, supposing a <y and < TT 8, we 

 have 



It follows from our hypothesis as to the continuity of /(.?) that, when any positive 

 number e is assigned, 'we may choose a so small that 



-/(*) KK 



for these values of s and t. Hence, observing that 



we obtain 



= 1 



mr Jo 



The second term on the right is not greater than ^e, and, since 



=0, 



we can clearly find a positive integer N, great enough to ensure that the first is not 

 greater than ^e, for all values of n 2: N,, and of s in (y, 8). Again, we can choose a 

 positive integer N a in such a way that the numerical value of the difference (38) is 

 less than ^c, for all values of n S N 2) and of s in (y, 8). It follows at once that, for 

 values of n which are not less than the greatest of N! and N 2 , we have 





for all values of s in (y, 8). 

 indefinitely, 



In other words, we have shown that, as n increases 



n 



converges uniformly to f(s) for these values of s. 



We may summarise the results obtained in this paragraph in the following 

 theorems : 



The arithmetic mean of the first n partial, sums of any canonical Sturm-Liouville 

 xi-ries corresponding to f(s) converges to lira ^[_f(s t) +y(s + <)], as n increases 



(-*-0 



inilijinitely^ at each point of the open interval (0, TT) where this limit exists as a finite 

 number; moreover, at a point where this limit has one of the improper values oo, it 

 diverges to this value and is non-oscillatory. If the set of points at which f(s) is 



