418 MR. J. MERCER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



The coefficient of F on the right has a sign opposite to that of the term independent 

 of k ; accordingly, when we equate the right-hand member to zero, the resulting 

 quadratic has its roots real. It follows that, if we suppose one of them to be a, the 

 function 



will satisfy (A), and it cannot be identically zero, because this would imply that 6 l (s) 

 is a constant multiple of 2 (s), and hence that the two integrals mentioned in (i) have 



the same sign. 



The converse of this theorem, however, is not true, for there are functions both of 

 the positive and of the negative type which agree in this property with those of 

 ambiguous type ; these are known as the semi-definite functions. The remainder are 

 called definite functions, and have the property that (A) can only be satisfied by a 

 function (x) which is zero at each point of (a, 1>). 



The two classes of functions we have just mentioned have distinctive properties in 

 the theory of integral equations. For, if K (x, t) is of positive or negative type, it is 

 evident from HJLBERT'S theorem that (A) can only hold when 



(*)0()cfo = (n= 1,2, ...) 



By a known theorem* we must, therefore, have 



' K (.s, t) (>) (It = (a < s < b). 



(I 



Thus the. nec-es*<u-tj diid sufficient condition t/iat a function of positive or negative 

 type sh-ould be definite is that it should be perfect. 



PART Tl. THK NATURE OF FUNCTIONS OF POSITIVE AND NEGATIVE TYPE. 



4. The double integral 



"('/<(, i) (x) (t) ds dt, ........ (1) 



- a J a 



in which K (s, t) is an assigned symmetric and continuous function, and 6 is any 

 member of the class B, may be regarded as the limit of a certain set of quadratic 

 expressions. For, let a,, 2 , ..., a n be points of the interval (a, b), taken in such a 

 way that the distances between consecutive members of the set of points consisting of 

 a, b and these n are all equal. Then, by the theory of double integration, and in 

 virtue of the symmetry of K (s, t), (1) is precisely equal to 



(b -a) 2 Lt [ K ( a i' 

 * 



n 2 



* Of. SCHMIDT, op. cit., pp. 451, 452, 



