15 6 OR- J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



periodic, the reader will be able to prove that it is also valid when s has either of the 



values v, ir. , c // \ 



Now, if ^) and o(.s) are the upper and lower bilateral limits of /(s) at any 



point s, we have 



( 14). Further, as it has been shown to hold for all values of s in the closed interval 

 (-TT, IT), (44) holds for any value of s whatever. We are thus in a position to state 

 theorems, analogous to those numbered I.-V. in the preceding paragraph, on the 

 behaviour of the FOURIER'S series (43). For example, corresponding- to III., we have 

 the theorem 



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

 FOURIER'S series cm-responding to /(s), if it converges, has this bilateral limit for its 

 sum. In particular, if the series converges at a point where 



exists, then this limit is its stim. 



We leave the reader to enunciate the other four theorems, merely remarking that 

 each of them is valid for unrestricted values of s.* 



22. From results which have been obtained above we may deduce theorems 

 concerned with the expansion of a function F(a-), which has a Lebesgue integral 

 in (a, b), as an infinite series of the type 



% (y) F (y) <*y +% (*) ff (y) ^ (y) F (y) d y 



., . . . (45)t 



where (x) is the solution of 



d fj dv\ , n 



j- (k -=- +(grl\ v = 0, 



dx\ dx) vy 



which, for r = r n , satisfies one of the pairs of boundary conditions B, a B, *B, *B. It is 

 assumed that the functions (x) are made definite} by imposing upon them the 

 conditions 



f E*.(*ff * - 1 (u=l,-2, ...), ...... (46) 



These theorems are, of course, more general than those obtained by the methods of FEJER and 

 LKBESGUK (ride HOBSON, "The Theory of Functions of a Real Variable,' pp. 707-714). 



It was explained in 1 that y is a function of z; we employ g(if) to denote the same function with the 

 argument y instead of x. 



There is an ambiguity of sign which, however, is of no consequence. 



