Mr Dixon, On a pi^operty of siimmahle functions. 213 



tends to zero with 6, if g is any positive number, and that the 

 same is true of 



F\f{x-ve)-f{x)\dx, 



J a 



where Fx is any function that is finite when x is finite and 

 tends to zero with x. 



3. To prove that 



/, 



f{t)f(t + x)dt, 



say (f) (x), is continuous, if (a, h), {a + x, h + x) are both included 

 in (a', h'), we have , , . 



(^ (^) - (^ {y) = f /(O [fit + ^) -fii + y)] dt, 



J a 



and therefore 



{<!> (x) - cf> {y)Y < f {f{t)Y dt r [fit + x) -fit + y)Y dt, 



J a J a 



in which the first factor is finite, and the second tends to zero with 

 X — y hj what has been proved. 



Hence (f)(x) — ^ (y) tends to zero with x — y, and (f) (x) is a 

 continuous function of x. 



If we now put a =—77, b = 7r and suppose /(^) to have the 

 period 27r, it is easy to find the Fourier expansion of (j) (x). 



First, (f) (x) is an even function of x. 



Secondly*, 



fir rir r-ir 



I <^ {x) cos nxdx = 1 I f(t)f{t + x) cos nxdtdx 



J —TT J —irJ —n 



f(i)f (y) COS n(y-t)dy dt 



J —TT J —IT 



— I /(O co^ '^^^dt X I f(y) cos nydy 



J —TT J —TT 



+ I f(t) sin ntdt x I fiy) sin nydy. 



J —TT J —IT 



Hence the Fourier constants of <f) (x) are 



7^ao^ TT (fti^ + 61^), TT {a.^ + b^^), . . . 

 0, 0, ... 



* In calculating I I ^/(« + .T)/(t) cos n.rdirf.T, we first change the order of 



integration, and this is jiastified for/(.x-) and f{x+y) are both summable in any 

 rectangle in the xy plane : the set for which k>f{x)>l consists of lines parallel to 

 the y axis and the set for which k>f(x + y)>-l of lines which make equal intercepts 

 on tlie axes. 



