126 OR- J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



when any positive number t is assigned, we can choose a positive number rj small 



enough to ensure that 



for all values of t in (a,, 6.) and for all pairs of points s+^s belonging to (a 2 , 6,) for 

 which \r)\<rj- Taking any fixed value of*, it follows from this that 



f 



J 



and hence that 



at each point * of (a a , b,). 



In the first place, let us suppose that /(s) is a limited function ; then 



x (s, t)f(t + s) 



is a limited function of t and has a Lebesgue integral in (a lt bj for each value of s 

 in (a,, &,). We therefore have 



t > . . . (10) 



for a, ^ s ^ b,. The right-hand member of this equation and each of the terms of the 

 series on the left may be shown to be continuous functions of s in the interval (a a , 6 a ). 

 For, supposing as above that s and s + r) both belong to ( 2 , fe a ), we have 



The first term on the right is not greater than 



where /is the upper limit of | /(*)); and the integral just written is not greater 

 than 





