Mr Dixon, On a property of summahle functions. 211 



This series is uniformly convergent for all real values of y, 

 and hence its sum is a continuous function of y ; putting x for y 

 we have that 



f(t)f{t + x)dt 



is a continuous function of x, if f(x) is any limited integrable 

 function, with the period 27r. 



The first object of the present note is to give a proof of this 

 theorem, independent of the Fourier theory. 



A special case is that 



f(t)f(t-^x)dt=r {f{t)Ydt 



J —IT 



{fit + x)Y dt, 



from which it follows that 



Lt r {f{t + x)-f{t)Ydt = 0. 



X^O J -IT 



This special case will be proved first, and the more general 

 theorem then derived from it. The method of proof is applied 

 to unlimited, as well as to limited, functions, and thus it is possible 

 to prove de la Vallee-Poussin's theorem for all cases in which it 

 has a meaning. 



2. Let f{x) be a limited summable function in an interval 

 {a', h') which includes a, h as internal points, and let U, L be its 

 upper and lower boundaries in {a, h). Divide the interval {L, U) 

 into n—\ equal parts, at a^, as ... «.«-!> and let a^= L, an = U. 



Let the set of values of x in (a, b) for which ar-i<f{oc) ^ a^ be 

 called er{r = l, 2 . . . %). 



Enclose e^ in a set of intervals A,., not overlapping, and the 

 complementary set 6^(e,.) in intervals F^, not overlapping, so that 

 Ay, F^ have a common part < a. This can be done, for f{x) is 

 summable. 



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

 Ki, ^n--- S'lid take an integer p, such that 



I S,,^>A,-e (r = l, 2...n). 



m = l 



Let er^m denote the part of hr,m which is also in F^. 



The similarly formed system of sets and intervals in 

 {a-\-6, h + 6), shifted back a distance 6, will serve for the function 

 f(x+ 6) ; let them be distinguished in this new position by dashes, 

 so that SV,m is S^.m ill the new position. Of course 6 <h' —h. 



