FORMAL REALIZARILITY THEORY — I 231 



VI. LINEAR CORRESPONDENCES 



(i.O 111 (l('\('l()|)itiu; llio formni pi-opcrtics oi" 'i/z-polcs \\lii(;h aro (M|iii\'alciit 

 to I he pli.N'sical ones just listed, it would be instructive to adjoin I'c- 

 (luiiciiKMits piccc'incal, iiuicli in the oi'der gix'en in Section 5. Space does 

 not permit us full enjoyment of this luxury, hut the reader will hud a 

 rough parallel between Section 5 and the developments of this Section 

 and Section 7. 



().l It is well known that linear time invariant systems are best studied 

 l)y the tools of Fourier or r^aplace analysis. We make this fact the basis 

 of our hrst step in characteriziiiju; physically realizable 2n-poles simply 

 by phrasing our whole discussion in the frequency language. The con- 

 tent of the following paragraph will be obvious enough, but it does de- 

 fine terms to be used later. 



6.11* Let V and /:, without underscores, represent ri-tuples of complex 

 numbers : 



V = [Vl ,V.2, ■■• , Vn], (1) 



k = [k,,h, ■■■ , k,,]. (2) 



These are to l)e manipulated by the rules of vector algebra. Let p be a 

 complex number. We shall say that a 2n-pole N admits the pair [v, k] 

 at freciuency p, if in the sense of 4.2 N admits the pair [ij, k] {with under- 

 scores) where y has components 



Vrit) = Re(y,e'^'), 1 < r < n, (3) 



and k has components 



kr{t) = Reihe'"'), 1 < r < n. (4) 



Also analogously to 4.2, we say that N admits v at frequency p if 

 there is a k such that N admits [v, k] at frequency p, and that this k 

 corresponds to v (at frequency p). Similarly, N admits A; at fretiuency p 

 if there is a (corresponding) v such that N admits [v, k] (at p). 



6.12* Let V denote the aggregate of all n-tuples (1), and K the aggre- 

 gate of all w-tuples (2). These are then complex linear spaces. 



6.2* As our first step toward characterizing realizable 2n-poles, let us 

 consider a linear correspondence L between V and K described by the 

 postulates : 



PI. There is a set Fz, of complex numbers and for each peF/, a list 

 L{p) of pairs [v, k], veY, fceK. 



* Technical paragni|)h as explained in Section 2.91. 



