I.Vj r>R. J. MEECER: STURM-LIOUVILLE SERIES OF NORMAL 



converges uniformly to zero, as X tends to - oo, for all pairs of values of s and t 



lying in the square O^S^JT, O^t^ir. 



When the pair of boundary conditions is ;B', (38) must be replaced by 



|X.-nJ<r; 



hence, in this case, the difference between (36) and 



* 3 \ T V/ T" \ / 



= i n A. 



converges uniformly to zero. 

 The reader will observe that corresponding changes must be made in the enunciation 



of the theorem of 18. 



20. Let i/r,(s), ^,(), .... */(), ... be a complete set of normal functions which 

 satisfy the differential equation 



-1-5 + (?+/*) w = > 



and any one of the pairs of boundary conditions B', B', "B', ;B' ; further, let the order 

 of these functions be such that the corresponding singular values increase with n. 

 Then, if f(s) is any function which has a Lebesgue integral in (0, IT), the terms of 

 the series 



will have a deBnite order, and the coefficients will each have a meaning. We shall 

 refer to (42) as a canonical Sturm- Liouville series corresponding to f(s). 



Let s be any point of the open interval (0, IT). Denoting by U (s) and L (s) the 

 upper and lower limits of indeterminacy of the series (42), the general theorem 

 of II., 4, shows that 



U 00 2: IbT -X [ K x (s, t) f(t) dt = Jim_ -X [ K x (s, t) f(t) dt>L (s). 



Also, by the results of 13, 19 we have 



^) 2: "Hm~-X [*K A (s, t)f(t) dt >: lim -X f'K, (s, t)f(t) dt^u (s). 



*-*- -o x-*-- Jo 



By supposing that o> (s) = (s), we obtain the theorem : 



I- U U() find L(s) are the upper and lower limits of indeterminacy at the point s 

 of one of the canonical Sturm-Liourille series corresponding tof(s), then 



U ()() * L(), . 



fit each point where u(s), tlie bilateral limit of f(s), exist*. 

 By supposing that U (*) = L (*), we see that : 



