154 DR. J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



the normal functions of the series satisfy the boundary condition u = at s = 0, for 



we then have 



U(0) = L(0) = 0, 



whatever be the nature of /(*). Similar remarks apply when s has the value v. 



21. The theorems of the preceding paragraph apply to FOURIER'S sine and cosine 

 series, since the latter are particular Sturm-Liouville series. It is not difficult to 

 extend them to the FOURIER'S series 



t f f(t) dt + - sin s f /(/) sin tdt + - cos s f f(t) cos t dt 

 '2ir ] w IT 1 * IT i w 



+ ... + - sin m \ f(t) sin nt dt + - cos ns \ /(/) cos nt dt + (43) 



7T .' -n' If 



corresponding to a function f(s) which has a Lebesgue integral in ( TT, IT), The 

 series just written is clearly 



1 f* If* 



+ ... + - sin ns\ [fit) f(t)^smntdt + - cosns [/(*)+ f( f )~\ cos nt dt+ .... 



L/\/ /\ /J L/ \ / */ \ / J 



IT ,'Q IT Jo 



Let us in the first place suppose that s is a point of the open interval (0, IT). It is 

 known* that the difference between the sums of the first n terms of FOURIER'S sine 

 and cosine series corresponding to [/(*)/(*)] converges to zero, as n is increased 

 indefinitely. Hence the limits of indeterminacy of (43) are identical with those of 



1 f 



+ - cos ns [f(t) +f(- 1)] cosntdt+...; 



IT Jo 



and, therefore, in virtue of the fact that the n lh term converges to zero as n tends 

 to oo, with those of 



- f f(t)dt + -coss ["/(<) cos / dt + ... + -cos ns f" f(t) cos ntdt + .. 



" IT 



Referring to the first inequality of 20, we deduce from this that U() and L(.v), 

 the upper and lower limits of indeterminacy of (43), satisfy the inequality 



, /(O * 2= L (s) ; 



* /We IV., 13-15. 



