FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 153 



IT. The sum of a canonical Sturm-Liouville series corresponding to f(s) at any 

 point where it converges lies between the upper and lower Inlateral limits of /(.) at 

 the point. 



Again, by taking the case in which U (s) = L (.), a(s) = o>(s), and the common 

 value is in each case finite, we have 



TTI. At any point win-re the. bilateral limit of f(s) exists as a finite number, no 

 canonical Sturm- Liouville series con'esponding to f(s) can be convergent and have its 

 sum different from this Inlateral limit. In particular, no canonical Sturm-Liouville 

 series corresponding to f(s) can converge and have its sum different from 



lim *[/(* + <) +/(-OJ 



t-fr-O 



at a point where this limit exists. 



This theorem may, of course, be regarded as included in I. or II. Another 

 particular case which is worthy of remark is that in which U (s) = L (s) = oo. 

 Since we can only have o> (s) = oo at a point of infinite discontinuity of f(s), we 

 see that 



IV. A canonical Sturm- Liouville series corresponding to f(s) can only diverge to 

 + oo, or to oo, and be non-oscillatory at a point of in/lnite discontinuity of f( s). 



Lastly, it is known* that |V(s)| is less than a fixed positive number for all values 

 of n and s. From the result of II., 7, we deduce that 



V. At any point where the bilateral limit of f(s) exists as a finite number, each 

 canonical Sturm- Liouville series corresponding to f(s) may be made to converge, to 

 tli is limit by the introduction of suitable brackets. 



Different systems of bracketing may have to be employed at the various points of 

 (0, TT), but, in virtue of results obtained later.t it will be seen that, at any particular 

 point s, the same system will suffice for each canonical Sturm-Liouville series. 



It will be observed that the above theorems have been stated only for values of s 

 in the open interval (0, TT). After what has been said in 15, 19, there will l>e no 

 difficulty in supplying the results which correspond to I.-V. when s is an end point 

 of the interval. It will be found that, if the boundary condition which is satisfied by 

 the normal function of the series is not u = 0, all the above results hold good for 

 s = 0, provided that we replace w(s) by/(0 + 0), &(.?) by/(0 + 0), and <a(s) by/(0 + 

 wherever necessary ; for example, corresponding to I., we have the inequality 



U (0) ==/(<) + 0) 2: L(0), . 



when f(s) is such that /(O + O) exists. It is unnecessary to consider the case when 



* <y. iv, G. 



t It is shown in the following section that, as n tends to oo, the difference between the sums of the 

 first n term* of any two canonical Sturm-Liouville series converges to zero at each point of the open 

 interval (0. IT). 



VOL. OCX I. A. X 



