210 Mr Dixon, On a property of summahle functions. 



On a property of summahle functioiis. By A. C. Dixon, Sc.D., 



Trinity College. 



[Received 22 March 1909.] 



[Read 3 May 1909.] 



1. It is a known theorem*, due to de la Vallee-Poussin, that 

 if f{x) is a limited integrable real function of x in the interval 

 (— TT, tt) and if 



IT'" 1 /''" 



ao=~l f{t)dt, an = -\ f(t) cos ntdt, 



1 f'^ 



hn = - \ f(t) sin 7itdt, 



IT J -^ 



then ^tto^ + 2 (a,j^ + b^^) is a convergent series, whose sum is 



It is also known that if a^, a/, ...,hi ... are the Fourier con- 

 stants of a second such function (f) {x), formed from it in the same 

 manner as a^, «!, ..., bi, ... from f(x), then 



^ tto fio' + 2 (a„ an + bn bn) 

 is a convergent series, whose sum is 



ir f{t)4>{t) 



dt. 



If we suppose /(^) to be periodic, with period 27r, and take 

 ^ (^) —f^^ + 2/)' '^® have 



1 f" 



/(i + 2/) cos ?iic?^ 



1 



/(^)cos 71 (^ — y^dt 



= a^ cos ny + &« sin ny, 

 and 6„' = 6„ cos ny — an sin ny similarly ; 



also ao = ao. 



Thus an an + bn bn = (a,i^ + &,i^) cos ny, 



1 /"IT OO 



and - f(t)f{t +y)dt = \ao^ + X (an' + bn') cos ny. 



"^ J -TT 1 



* For proof and references see Hobson, Functions of a Real Variable, pp. 715-7, 

 723-5 ; B6eher, Annals of Mathematics, ser. 2, vol. vii. p. 107. 



