BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 527 



for the unit matrix (5^) in which da = 1, 5y- = for i 9^ j, and A~^(x) 

 stands for the matrix inverse to A{x). The matrices F'^{x), 6r+(x), 

 F~(x), G~{x) are to be determined to satisfy (11), the first two as 

 matrices of regular inner functions and the last two as matrices of 

 regular outer functions. The matrix products on the right hand side 

 are the customary matrix products, and the factors r^ (x), t~'p(x) stand 

 for the matrices (r^ (x) 5^) and (r-^ (x) 8ij) respectively. 



It may be proved without difficulty that for p taken large enough 

 a solution of these equations exists. To effect this proof we apply a 

 method of successive approximations based on the sequence of 

 equations 



FoHx) = F,-{x) = Go'ix) = 0, Go^ix) = I, 

 i Fi+ (.t) - Fr (x) = T^ (x) GV (x) A (x), 

 (12) I Gr (x) - G,^ (.r) = T-P (x) Fo- (x) A-' (x) - I, 

 ( F2+ (x) - F,- {x) = rP (x) G/ (x) A (x), 

 i Gi- (x) - G,^ (x) = r-P (x) Fr (x) A-' (x) - I, 



The symbol is used to denote a matrix of zero elements, and the 

 superscripts + and — are used to designate matrices of regular inner 

 and outer functions respectively. 



If we write Po-{x) = Fo'ix) = 0, QoK^') = Go+ix) = I, and further- 

 more 



along C, 

 along C, 



(13) 



\ Pro" {X)'= Fm+ (x) - Fm-iHx), Pm' (x) = FrrT {x) - Fm-r {x) , 



\ Qm" {X) = Gr,,^ (X) - Gm-l^ (x), Q^- (x) = GrrT {x) - Gm-l' (x) , 



it is clear that the sequence of equations (12) is equivalent to 



I Qm-{X)-Qm*(x) = r-P(x) P^-^-(x) A-Kx), (^ = 1, 2, . . .). 



Here the superscripts are employed as before. 



The form of equations (14) is such that we can determine P^^ (x), 

 Pm~{x), Qm^{x), Qm~(x) in terms of Pm-i'ix), Qm-i'^(x) so that the 

 mth pair of equations (14) is satisfied. In fact the first one of the 

 mih pair of matrix equations may be broken up into n^ ordinary 

 equations 



n 



Pi.j.m'^ix) — Pi.j,m~(x) = t'p {x) 21 qi.\,m-\'{x)a-Kj {x) along C, 



x=i 



{i,j = 1, . . .,n), 



