520 SCIENCE PROGRESS 



o„ fl, a, a^a.a^ . . . should be different from zero. If the 



a's are to be integers, these determinants A„ will also be integers, 

 and it will simply be necessary to have Urn A„ = O. It will be 



rt-> 00 



noticed that the same would be true if the a's were integers in 

 an imaginary quadratic field, and the word " integer " can be 

 taken in this extended sense in the theorem. The next step is 

 the introduction of the Tschebyscheff pol^^nomials, which for a 

 given degree differ as little as possible from zero in a given 

 region. Faber {Crelle, 150, 1920, 70-106) showed how con- 

 formal representation could be used to find the order of magni- 

 tude of such polynomials, and it is thus that conformal repre- 

 sentation comes into the theorem. 



AlHed to this work is that of G. Szego {Math. Ann., 87, 1922, 

 90-1 1 1 ; Sitzimgsber. Berlin, 1922, 88-91) on power series with 

 only a finite number of different coefficients. Such a series 

 either represents a rational function or is not continuable beyond 

 unit circle. In the first case the coefficients «„, from a certain 

 n on, are periodic and the function is equal to a polynomial 

 divided by i - z^. See also a paper by Ostrowski {Sitz. 

 Berlin, 1921, 557-65) on power series with an infinite number 

 of vanishing coefficients. 



The zeros of the Tschebyscheff polynomials, and other poly- 

 nomials which like them arise from minimum conditions, have 

 been investigated by L. Fejer {Math. Ann., 85, 1922, 41-8). 

 He starts from the proposition that if we have a point set P 

 in the finite part of a plane, which is either closed or consists 

 of a finite number of points, we can always find a point B 

 which is nearer to any point of P than A is to the same point, pro- 

 vided that A lies outside the convex envelope of P. If gn{p) 

 is the value of a polynomial of the nth degree at any point p 

 of P, then \gn{P)\ has a maximum which may be called the 

 T-divergence (Abweichung, ecart). The Tschebyscheff poly- 

 nomial of degree n for P is defined by the inequality 

 Max \ln{p)\ < Max \grip)\ ; and Fejer proves that the zeros of T„ 

 lie in the convex envelope of P, provided that P contains at 

 least n points. As a particular case, if P is the straight line- 

 segment - I iC ::k: ^ I , then the roots are all real and in this in- 

 terval. Other types of divergence may be defined to give other 

 polynomials, for example, the Bessel-divergence, defined for a 

 finite number of points as {^(A)' + • • • + g{pky}/k, or for the 



I 



points of a rectifiable curve of length / as j \g{P)\'<^s, and similar 



conclusions are arrived at for the zeros. An extension of some 

 of this work is given by M. Fekete and J. L. v. Neumann 



