192 



DR. J. MERCER: STURM LIOUVILLE SERIES OF NORMAL 



I to /'(*) wiil'-rnihi in tli!* interval. 

 If the normal functions of a Sturm- Liouville series do not satisfy the boundary 



condition u = at S " , then, as n increases indefinitely, the arithmetic mean of the 



8 7T - , ^ 



fir*, n partial mu of the series convenes at S g = to f /Jf whene r this limit 

 exists as a finite number; moreover, when this limit has one of the improper valves 

 oo, the arithmetic mean diverges to this value and is non-oscillatory. 



22. We proceed to apply the foregoing results to the more general Sturm-Liouville 

 series & 



*, (*) f .</ (y) *i (y) F (y) d y + * (^ ( a (y) *> (y) F 



.(y)F(y)^+ ...... ( 39 ) 



We saw above (III., 22) that the terms of this series are identical with those of 



^ (40) 



"/ ^0 '" \"/ " ^ 



where s is the point of (0, IT) corresponding to the point x of (a, b). 

 In connection with the series just written, let us consider the series 



. . (41) 



This latter is a canonical Sturm-Liouville series corresponding to f(s) ; and/(s), it 

 will l)e recalled, is F (.r) expressed as a function of s. We shall refer to (41) as the 

 canonical Stnrm- Liouville series related to the general Sturm-Liouville series (39). 



Let a-, (s) be the sum of the first n terms of the series (41) ; and let & n (s) be the 

 sum of the first n terms of the series (40). We proceed to show that, as n increases 

 indefinitely, 



<r B (a)-or.(s) 



converges to zero uniformly in the whole of (0, IT). 



In the first place, let us suppose that the pair of boundary conditions satisfied by 

 the functions (x) is B ; the normal functions \jt n (s) will therefore satisfy B'. By 

 the result obtained in 10, it is known that, as n increases indefinitely, 



converges uniformly to zero in (0, IT); and by the results obtained in 13, 14 (c;/ 1 . 21 ) 



^ii^-i)^ dt 



Sill TJ-* 



