PROCEEDINGS 



OF THE 



On Certain Trigonometrical Series which have a Necessary and 

 Suffijcient Condition for Uniform Convergence. By A. E. Jolliffe. 



(Communicated by Mr G. H. Hardy.) 



[Received 1 June 1918; read 28 October 1918.] 



1. The series S^n sin nd, where (a,i) is a sequence decreasing 

 steadily to zero, is convergent for all real values of 6, and it has 

 been proved by Mr T. W. Chaundy and myself* that the series is 

 uniformly convergent throughout any interval if /?a,i-*-0, this con- 

 dition being necessary as well as sufficient. 



A generalization of this theorem is as follows : 



If (Xn) is tt sequence increasing steadily to infinity and (an) is 

 a sequence decreasing steadily to zero, then the necessary and suffi- 

 cient condition that the series Sa„+i(cos A,.,i^ — cosX,i_^i^)/^, which is 

 coyiver gent for all real values of 6, shoidd be uniformly convergent, 

 throughoid any interval of values of 6, is Xnan^b. 



I shall prove rather more than this, viz. that the condition is 

 sufficient for uniform convergence and necessary for continuity. 



When ^ = 0, it is understood that the value assigned to any 

 term of the series is its limit as 6 tends to zero, so that for ^ = 

 the sum of the series, which I shall denote by Sun, is zero. Since, 

 by Abel's lemma, 



i Wn+i + ... + Up\< 2an+i/0, 



it is evident that there is continuity and uniform convergence 

 throughout any interval which does not include ^ = 0, so that it 

 is only intervals which include ^ = that we have to consider. 



* Proc. London Math. Soc. (2), Vol. 15, p. 214. 

 VOL. XIX. PART V. 14 



