214 Mr Dixon, On a property of summahle functions. 

 if those oi f{x) are 



tty, Cll, di^y ..., 



61, h, .... 



n If'" 



Since ifto' + t {an"" + K^) < - {/(^)}' ^^ 



1 TT J -n 



for all values of w, the Fourier series is absolutely and uniformly 

 convergent, and since (}){x) is continuous, the sum of the series is 

 equal to (f) (x), that is, 



(f){x)= I irao^ + TT 2 (ttn^ + bn^) cos nx. 

 1 



(Hobson, F. R. V., p. 713.) 



4. Suppose now that f(x) is not a limited function, but is 

 still summable in an interval (a', h'), which includes a, h as 

 internal points. It will be proved that 



\\f{x + e)-f{x)Ydx 



J a 



tends to zero with 6, if I \f{x)Y dx exists. 



J a 



Take a finite quantity h, which will afterwards be made to 

 diminish without limit. Let er{r = 0, ± 1, ± 2 ...) denote the set 

 of points in (a, h) where {r—l)h <f(x)^rh. 



Take a whole number n, so that I {/ (^)j^ dx, taken over the 



set complementary to ei^n + ••■ +&0 + S1+ ... +en, is < 7. 



Enclose e^ in a set of intervals A,., not overlapping, and C(e,.) 

 in intervals F^, not overlapping, so that A^, F^ have a common 

 part not exceeding a/(2?^+l)^. The sum of these common parts 

 for all values of r is then < 3a. Then we shall have 



Ao+Ai + ...+A„ 

 + A_i + . . . + A_„ > (6 - a) - /3, where /3 = y/n^'h^ 



Let the intervals of A^, in descending order of length, be 

 Sn, 8r2". and take a whole number p, such that 



Z 8r,m>^r-e (r = 0, ±1, ...±n). 



Let er^m be the part of h^^m which is also in F,., and let the 

 dash again indicate a displacement Q to the left. 



Then for any x which is common to hr^m and SV,m and is not 

 in F,. (jr F',., 



\f(^^^e)-f{x)\<K 



