56 



Consider x^"" where m is any positive integer; and let stand for the 



continuous function given by 



1 



Then it is easily seen that 



where 



0 



— 2m (2m - 1) . . . (2i- + 1) l^^"" 5, , r = - 1, - 2, ... 1, 0, 

 the B's being given by the simultaneous equations 



(2w + l-2s)! ' (2/U-1-2S)! ' ' (2r + l-2s)! ' ^ 1! 

 s = w — 1, m — 2, . , . 1,0. 



Similarly, if 



^2™+i _ 1 r'^'^^c^ (I) l < it; < TT - l , 

 a;— 1 



where 



6,. = 1, r = m, 



= (2m + 1) 2m . . . (2r + 2) l^^"*-»") jj^^ r = m - 1, m -2, . . . , 1, 0. 



Now, according to a well-known theorem due to Weierstrass it is always 

 possible, by taking n sufficiently large, to find a rational and integral polynomial 

 (x), of degree n, such that 



I T, (x) ~ P„ I < (J, 0 < it; < :t. 



Therefore, if 



n n—1 n , n — 1 



0 0 



the required admissible characteristic function is 



11 11—1 n n — 1 



c (1, 0) = s j 2 1 + s i s r-^' I . 



1) See Weierstrass's Memoir, loc. cit. p. 796. 



