1910 - 11 .] 
561 
On Fourier’s Repeated Integral. 
(ii.) for some values of the constants C + 
and C_ , 
f’MzQd u and r M=°rdu both 
J 0 U J -q U 
exist as Lebesgue integrals; the right-hand 
side of the above equation then becomes 
f(C + + C_); 
or 
(iii.) f(u) is expressible in an interval con- 
taining the origin as internal point , as the 
product g(u)h(u) of two functions , one of 
which, g(u), satisfies (i.), and the other , h(u), 
satisfies (ii.). The right-hand side of the 
above equation must then be suitably altered 
in accordance with (ii.). 
(iv.) the repeated integral exists ; f(u) has a 
Lebesgue integral in some interval ( - p, p), 
and the origin is a point at which f(u) is the 
differential coefficient of its indefinite integral. 
Alternative Conditions at Infinity. 
(I.) J* | f(u) | du and | f(u) | du exist, or, more generally , J“| f(u) + f( -u) | du 
exists, for some value of the positive quantity q ; or 
(ii.) in some interval (q, oo ), f(u) = g(u)h(u), where g(u) is monotone decreasing 
with zero as limit at infinity , while h(u) is any summable function whose Lebesgue 
integral can be expanded in a series of the form 
r 00 . ' 
yh(u)du = A 0 u + 5(A n cos k n u + sin k n u), 
the positive quantities k n increasing without limit as n increases, and the series whose 
general term is | A n | + | B n I converging, while the integrals I ^ll)du and I 
J q ll J 
M 
u 
du 
both exist. 
It is hardly necessary here to point out that f(u) may, in any one of 
the parts into which we thus divide the straight line, be the sum of any 
finite number of functions, satisfying the above conditions, each multiplied 
by a suitable constant. In particular, we may take g(u) in (II.) to be a 
function of bounded variation in • the whole infinite interval, instead 
of monotone decreasing, provided each of the monotone descending 
functions of which it is the difference satisfies the further conditions 
given. 
There are various special cases of the condition (II.) which are of greater 
interest than the general case. 
VOL. XXXI. 
36 
