MATHEMATICS 175 



T. Carleman {C.R., 174, 1922, 1527) finds conditions under 

 which a function is determined uniquely by its asymptotic 

 series. 



G. Mignosi {Rend. Napoli, 27, 1921, 17-28) and M. Cipolla 

 {ibid., 28-37) give criteria for convergence of series related 

 to those of Hardy ; and C. Severini {Rend. Lincei, 31, 1922, 

 97-101) gives criteria for the uniform convergence of a sequence 

 convergent in media. 



Completing the work of Helly and Bray, T. H. Hilde- 

 brandt {Bull. Amer. Math. S., 28, 1922, 53-8) finds necessary 

 and sufficient conditions for the convergence of sequences of 

 linear operations. 



L. Bieberbach {Math. Ann., 85, 1922, 141-8) writes on the 

 distribution of the points at which an analytic function takes 

 the values zero or unity; E. Landau {ibid., 158-60) shortens 

 his work. Allied questions are dealt with by P. Montel {C.R., 

 174, 1922, 22, 143 ; Mem. soc. roy. d. Belgique, 6, 1922, fasc. 15) 

 in his papers on quasi-normal families of functions ; see also 

 T. Varopoulos {C.R., 174, 1922, 272). 



S. Sarantopoulos {C.R., 174, 1922, 591) and P. Montel 

 {ibid., 850, 1220) give theorems fixing an upper limit to the 

 moduli of zeros of polynomials subject to certain conditions. 



If a polynomial have only real zeros, all its derivatives have 

 only real zeros ; G. Polya {Math. Zs., 12, 1922, 36-60) examines 

 other classes of functions with this property. 



As far back as 1 894, Borel introduced a class of functions, 

 called quasi-analytic, for which the Taylor expansion diverges 

 at every point, but which are entirely determined by the 

 knowledge of their derivatives at a point. A. Denjoy {C.R., 

 173, 1 92 1, 1329) fills a gap in the theory by giving simple 

 criteria for such quasi-analytic properties, and, as a result, 

 several notes on the subject have appeared : E. Borel {C.R., 

 173, 1 92 1, 143 1 ; 174, 1922, 505), M. Gevrey {ibid., 368), 

 G. Juha {ibid., 370), T, Carleman {ibid., 373, 994). 



J. Wolff {C.R., 173, 1921, 1327), A. Denjoy {C.R., 174, 1922, 

 95) and T. Carleman {ibid., 588) discuss functions defined by 

 series of rational fractions. 



E. H. Moore {Math. Ann., 86, 1922, 30-9) gives a generalisa- 

 tion of power series in a denumerably infinite system of 

 variables. 



Various papers on the singularities of power series and 

 Dirichlet series are by O. Szasz {Math. Ann., 85, 1922, 99-110), 

 by G. Szego {Sitz. Berlin, 1922, 88-91) ; by G. Polya {Proa. 

 L.M.S., 21, 1922, 22-38) on series with integral coefficients ; 

 by F. Carlson and E. Landau {Gottingen Nach., 192 1, 184-8) 

 and L. Neder {Math. Ann., 85, 1922, 111-14) on the extension 

 of Fabry's gap-theoremi 



