Proof of three propositions of Paley

where 4̂ > 0 and Y > 0 . The role of this estimate was that of assuring a high degree of smoothness for the distribution function having the infinite product as Fourier-Stieltjes transform. In fact, this distribution function is an entire function if 7 > 1 , it is regular analytic if y = 1 and it has derivatives of arbitrarily high order whenever there exists a 7 ( > 0 ) . For the sequences {an} in question and also for some more general sequences a simple proof of (1) will be given by means of an appraisal indicated in the reference cited. This appraisal has the advantage of being valid for every {an} and it gives results also if (1) is not satisfied. I t implies for instance the fact that the infinite product —»0 as t—» + oo in cases of the type an = c~ n , where c>l and 0 < 5 < 1 . The nth. factor of (1) does not come near to ± 1 if 1 ^aJS2f since this limitation implies |cos (ant)\ < 2 / 3 . Thus