416 ME. J. MEECEE: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



PART I. DEFINITIONS AND DEDUCTIONS FROM HILBERT'S THEOREM. 



1. Let K(S, t) be a continuous symmetric function of the variables s,t which is 

 defined in the closed square a < s ^ b, a < t : b ; and let <s> be the class of all functions 

 which are continuous in the closed interval (a, l>). When the function ranges 

 through the class , there are three possible ways in which the double integral 



\ K (s,t)e(s)6(t)dsdt. 



J a J a 



may behave : 



(i) There may be two members of 6, say t and 6 2 , such that 



f f K (x, t) 6, (s) e i (t) ds dt, \" l K (s, t) 6, (s) 2 (t) ds dt 



J a J a J a J a 



have opposite signs ; 



(ii) Each function may be such that 



f f K (s,t)0()8(t)d#dt>Q; 



* a " 'a 



(iii) Each function may be such that 



K (s, t) 6 (s) 9 (t.) <!x (It < 0. 



This suggests a classification of continuous symmetric functions defined in the 

 closed square. We shall speak of those which have the property (i) as functions of 

 ambiguous type, whilst the others will be said to be of positive or negative type, 

 according as they satisfy (ii) or (iii). 



2. From the point of view of integral equations this classification is of considerable 

 importance. HILBERT has proved* that 



\" f K (s, t) (s) (0 ds dt = S 1 [7 ^ (*) (s) 



la- a = ] A n |_* a 



where ^ (s), t// 2 (), ..., />($), ..., are a complete system of normal functions relating to 

 the characteristic function K (s, t) of the integral equation 



and \i, X 2 , ..., X n , ..., respectively, are the corresponding singular values. It follows 

 at once from this that, when the singular values are all positive, K (s, t) is of positive 



* 'Gott. Nachr.' (1904), pp. 69-70. See also SCHMIDT, 'Math. Ann.,' Band 63, pp. 452, 453. We 

 shall refer to the result given above as HILBERT'S theorem. The theorem stated by HILBERT on p. 70 of 

 the paper referred to can be deduced by writing (s) = x(s) + y (s) in the equation written above. 



