658 
MATHEMATICS: G. D. BIRKHOFF 
in which is a continuous functional of x and g{x), bounded as long as 
g{x) and dg{x)/dx are bounded uniformly for all n. 
For the satisfactory investigation of the representation of an arbitrary 
function in Sturm-Li ouville series it is necessary to know further that 
the set Ui{x) , U2{x) , . . . thus obtained is closed. A very simple proof 
and the first is due to Stekloff.^ His proof uses, however, the theory of 
functions of a complex variable. There is indeed a special case, namely 
the case when g{x) vanishes identically and u„(x) = \/2 sin mrx, 
when a short proof is possible by an elementary method. 
The theorem proved above makes possible an immediate demonstra- 
tion that the set is closed, once the asymptotic formula above and the 
fact for the special case are granted. Let us replace g{x) by (rg(x) where 
0- is a real parameter varying from 0 to 1. If then Ui{x), u^ix), . . . 
denote the members of the set for any particular o- and U\{x), W2(x), . . . 
denote the members for another value of o-, say cr, it will suffice to show 
that the series for H will have the properties demanded in the theorem 
for I (7 — 0- I ^ 5 > 0. For in this event since the set is closed for o- =0 
(the special case), it will be closed for o- ^ 5 by an application of the theo- 
rem ; by successive further applications of the theorem it is shown in the 
same way that the set is closed for o- ^ 25, o- ^ 35, . . . , so that finally 
we infer that the theorem is true for o- =1, i.e. for the given set. 
In virtue of the explicit formula for w„ {x) we have 
«„ {x) - «„ (*) = V2 [(^^il^Mf^il^? + M,{x,.g{x))- M„{xrag{xm 
L n fr J 
r \ /^r • I <p(y) cos niry . M {y,(Tg(x))l 
L n n A 
Multiplying these two expressions on the right together we obtain the 
typical term of the B. series. This series breaks up at once into six 
(2 X 3) other series all of which converge uniformly to a small value 
for 1 0- — (7 1 small save the series obtained from the leading terms in 
both expressions, namely 
2 ((7 - (t) <^ {x) 2 
cos n-KX sin niry 
n 
If we omit the small factor (a — u) ip (x) , this series may be written 
>^ sin n-K (x -f >') — sin n-w {% — y^ 
