1910-11.] 
On Fourier’s Repeated Integral. 
583 
g(u) cos uv has bounded variation, and h(u ) is summable, since ( h(u )) 2 is 
summable. Therefore * we may integrate the product term-by-term, 
using the not necessarily convergent Fourier series of h(u). Thus 
jqfl(u)h(u) cos uv du= % cos uv(a n cos nu + b n sin nu)du . . (1 ) 
Let m be the first integer A 2B, so that, for all integers n\m , 
n - B 
For such values of n, provided 0 L v L B, 
n + v \ n -v \ \n, 
Therefore, expressing the integrand of the integral under the summation 
sign in (1) in the usual way as a sum of sines and cosines, after removing 
the factor g(u) outside the integral sign, by the Second Theorem of the 
Mean, we get, for n\m , 
/: 
g(u) cos uv(a n cos nu + b n sin nu)du L g(q) 
1 1 
n + v n - v. 
L ±g(q){ \a n \ + \ b n | )/n 
( | a n | + I K | ) 
( 2 ) 
Now, denoting the sum of the first 2(m— 1) terms of the Fourier series 
of h(u) by s m , we may write the equation (1) in the following form : — 
co 
\ Q g(u) cos uv{h(u ) - s m }du = 2$ \^g(u) cos uv{a n cos nu + b n sin nu\du. 
The inequality (2) shows that we may apply the lemma of § 15 to the 
infinite series on the right-hand side of this equation, bearing in mind that 
the individual integrals in this series continue to exist when we write 
infinity for Q. Thus we can assert that both sides of the last equation 
continue to exist when we write infinity for Q, and that the equation still 
holds when we do this. Now, when this has been done, the right-hand 
side converges boundedly for all values of v in the closed interval (0, B). 
Hence we may integrate term-by-term with respect to v, and assert that 
[ B dv g{u) cos uv{h(u) - s m }du = 5 f ^g^) cos uv{a n cos nu + b n sin nu}du, (3) 
so that both sides exist and are equal. 
Again, since g(u)/u is monotone decreasing, and sin B u may evidently 
take the place of cos uv in the preceding argument, 
I s in B u[h(u) — s m ]du = 2, j g j n cos nu _|_ b n sin nu}du ; 
and, by the same argument as that used above, this equation has a definite 
* Loc. cit., supra , § 17. 
