186 



DK. J. MKRCKR: STURM-LIOUVILLE SERIES OF NORMAL 



point (s) belonging to the open interval (0, TT), then all the canonical Sturm-Liouville 

 series corresponding to /() converge, and have the sum J[/(*-0) + /( + 0)], a< t*M 



Again, it is easy to see that the above condition is satisfied when 



has a Lebesgue integral with respect to t in an interval (0, a). For, in this case, 



/(*-<) +./l(*+0-M) 



also has a Lebesgue integral in the interval, so that 



- 



dt, 



where, by a known property of Lebesgue integrals, the right-hand member tends to 

 zero with ft. 



Since fi(st) may be replaced by/(*<) for values of a which are sufficiently 

 small, we have thus established the theorem : 



At any point s of the open interval (0, TT), where o> (s), the bilateral limit of f(s), 

 exists as a finite number, a sufficient condition that all the canonical Sturm-Liouville 

 series corresponding to f(s) may converge, and therefore have a> (s) for their common 

 sum, is that 



should liave a Lebesgue integral with respect to t in an interval (0, a). 



In particular we have the following corollary : 



At any point s of the open interval (0, TT), ivhere f(sQ) and f(s+0) exist as finite 

 numbers, a sufficient condition that all the canonical Sturm-Liouville series corre- 

 sponding to f(s) may converge, and therefore have |-[/(s 0) +/(s + 0)] for their 

 common sum, is that 



should both possess Lebesgue integrals in an interval (0, a).* 



19. Consider next the case in which s has the value zero. It follows from what 

 was said in 17 that, with the exception of those whose normal functions satisfy 

 the boundary condition u = at this point, all canonical Sturm-Liouville series 



For an amplification of these theorems c/. HOBSON, 'The Theory of Functions of a Real Variable,' 

 pp. 60-683. 



