FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 159 



Denoting by fi (x) and fl(r) the upper and lower bilateral limits of F (x) at the 

 point x, and employing the notation of 13, it follows from (50) that 



w(x) ^ (l(x), u(s) S n (x). 



Again, if we commence with any two functions 9 t (y), 9g(y) having the properties 

 above mentioned, it is easily seen that the relations (51) together with t = k~ l '*g l 'y 

 define \i (0- X(0 as functions of t satisfying the requirements laid down in 11, 12. 

 We now deduce from (50) the relations 



s n (x), <a(s) i n(x). 



It follows that we must have 



f 



oTp) = n (x), at (a) = fl(x). 



25. From this result, and the inequalities established in 13, we infer that 

 ^>: lim'-X [*Ki(8,t)f(t)dt^ lim - 



at any point of the open interval (a, b). Taken in conjunction with the inequalities 

 (49), these at once lead to theorems on the convergence of the general Sturm- 

 Liouville series, corresponding exactly to those numbered I.-V. in 20. We shall 

 therefore content ourselves with the enunciation of that corresponding to III. This 

 reads as follows : 



At any point where the bilateral limit of F (x) exists as a finite number, no Sturm- 

 Liouville series corresponding to F (x) can be convergent and have its sum different 

 from this bilateral limit. In particular, no Sturm- Liouville series corresponding to 

 F (x) can converge and have its sum different from 



y-*-0 



at a point where this limit exists. 



As regards the end points of (a, b), we clearly have 



F(a+0)=/(0 + 0), F(a + 0)=/(0 + 0), 



F(6-0) =/(-0), F(fc-O) = /(TT-O). 

 Hence, referring to the results of 15, 19, we see that 

 F(a+0)==TmT-XrK,(0, t)f(t)dt=z lim -x['K A (0. t)f(t)dt 



**-- Jo *- -o. J 



when the boundary condition at x = a is 



