FUNCTIONS IN THE THEORY >K INTKCKAL EQUATIONS. 189 



A avjh'i-K'i't fiiiiilition that all the canonical Sturm-Liouville series corresponding/ to 

 f(s) may converge uniformly in an interval (y, 8) lyiny within (0, IT) is that 



ft ff t\j- /Yo.L.A_v> t'U\ 



dt 



should exist as a Lebesyue integral for each value of s in (y, 8), and that as ft 

 <Ii HI in idics indefinitely, it should converye uniformly to zero in (y, 8). In particular, 

 it is sufficient that 



I 



, 



f(s-t)-f(s) 



.' 



should both exist, and converye uniformly to zero.* 



21. Let s be a point belonging to any interval (y, 8) Avhich is contained within 

 (0, TT). Let cr B (*) be the sum of the first n terms of any canonical Sturm-Liouville 

 series at the point s, and let s(s) have the signification of 10. It follows from the 

 results obtained in 10-15 that, as n increases indefinitely, 



converges uniformly to zero in (y, 8). Hence, in virtue of the lemma of II., 10, we 

 see that 



o-i (a) + <r, ()+... + <r.() ?i () + *, (s) + ... + ?,(*) 



n 



n 



converges uniformly to zero in this interval. In particular, since (y, 8) is any interval 

 contained within (0, TT), we see that this difference converges to zero at each point of 

 the open interval (0,7r). It will be seen from the results of 10-15 that this is also 

 true at an end point, provided that the normal functions of the Sturm-Liouville series 

 do not all vanish there. 

 Now we have ( 13, 14) 



win-re F(t) = 



. Hence we obtain 



n 



* It is well known that, as increases indefinitely, the integral 



j; [/,(.->+/, c+o-s/wi^gj^* 



converges uniformly to zero for values of .< in (y, 8), if, sis / diminishes indefinitely, 



uniformly to zero in an interval miitaining (y, S) in its interior (vide HOIISON, 'The Theory of 

 Functions of a Ueal Variable,' pp. 691-694). This gives another condition for the uniform convergence of 

 Sturm I.iimville series. The formal statement of it is left to the reader. 



