FI'VCTIONS IN THE THKORY OF INTEGRAL EQUATIONS. 



197 



finite numbers, a sufficient condition that the Sturm- Liouville series (39) may converge, 

 and therefore have i[F(x 0) + F(.i* + 0)]/or its sum, is that 



y 







should both hare Lebes</ne integrals with respect to y in an interval (0, a). 



There is no necessity to give the analogues of the theorems of 19, 20, for their 

 statement can present no difficulty. The reader should observe, however, that, 

 corresponding to the particular case mentioned in the enunciation of the last theorem 

 of 20, we have a criterion of uniform convergence which does not involve the functions 

 7/1 and y a . It is, as follows : 



A sufficient condition that the Sturm,- Liouville series (39) may converge uniformly 

 in an interval (a,, &,) lying within (a, b) is that 



F(-y)-F(s) 



y 



, f> 



y 



. 



should exist as Lebesr/ue integrals, for each value of x in (a,, 6,), and that, as ft 

 diminishes indefinitely, both should converge uniformly to zero in (a,, &,). 

 26. We saw in 23 that the difference 



<r. (*)-?.(*) 



converges uniformly to zero in the whole of (0, IT). From the lemma of II., 10, it 

 follows that the difference between the arithmetic mean of the first n partial sums of 

 the series 



^ (40) 



and the arithmetic mean of the first n partial sums of 



*. (*) r * (o/(o *+*.() r * 



*o Jo 



converges uniformly to zero, as n increases indefinitely. 



Recalling that the terms of (40) are equal to the corresponding terms of 



*. (*) f 9 (y) *. (y) F (y) dy+V, (x) f ,/ (y) V, (y) F (y) dy 



(39) 



and applying results proved above (21) in regard to the convergence of the 

 arithmetic mean of the first n partial sums of (41), we obtain the theorems : 



The arithmetic mean of the first n partial siims of the Sturm-Liouville series (39) 



