1910-11.] SommerfehTs Form of Fourier’s Repeated Integrals. 603 
§ 15. Theorem. — If in the interval (q, oo ) the function f(u) can be 
expressed as the product of two factors , one of which, g(u), is monotone 
decreasing with zero as limit at infinity, and the other, h(u), is any periodic 
function whose square is summable, then 
L t f dv ( e~ kvH f(u) sin uv du= [ ^~—du , 
t=oJ 0 J q J q U 
provided the latter integral exists. 
The proof of this theorem is word for word the same as that in § 8, 
using the lemma of § 12 instead of that of § 6 to reverse the order of 
integration at the beginning. 
§ 16. It remains only to point out that the lemma of § 13, and, con- 
sequently the theorems of §§ 13 and 14, remain true when we remove the 
restriction that the square of h(u) is summable, provided only h(u) is 
summable and the series whose general terms are | a n \ / n and \b n \ / n are 
absolutely convergent. The reasoning of the text is, in fact, unaffected. 
Still more generally, reverting to the general theorem on the integra- 
tion of series * which forms the basis of the theorem on the integration of 
Fourier series quoted in § 12, we may replace the integer n by any positive 
quantity h n , having, as n increases, the unique limit infinity, and \a n \ /n 
and | b n \ fn by I A n | and | B„ | , pro vided h(u) is summable and its Lebesgue 
integral can be expanded in a series of the new form, viz. 
h{u)d,u = A 0 w -f % ( A n cos k n u + B w sin k n u). 
n—\ 
* W. H. Young, “ On the Theory of the Application of Expansions to Definite Integrals, 5 ’ 
1910, presented to the London Mathematical Society. 
{Issued separately July 26, 1911.) 
