Functions of positive and negative type, and their connection with the theory of integral equations

This memoir is concerned with continuous symmetric functions k ( s, t ) for which the double integral ∫ a b ∫ a b k ( s, t ) θ ( s ) θ ( t ) dsdt is either not negative, or not positive, for each function θ( s ) which is continuous in the interval ( a, b ) ; in the former case the function k ( s, t ) is said to be of positive type, while in the latter it is said to be of negative type. The importance of these classes of functions in the theory of integral equations will be gathered from Part I. The greater portion of the second part is devoted to a proof of the theorem that the necessary and sufficient condition, under which a continuous symmetric function, k ( s, t ), is of positive type, is that the functions k ( s 1 , s 1 ), k ( s 1 , s 2 s 1 , s 2 ),....., k ( s 1 , s 2 ...., s n s 1 , s 2 , ......, s n ), ... should never be negative, when the variable s 1 , s 2 , ...., s n , .... are each confined to the closed interval ( a, b ). This leads to several interesting properties of such a function ; for instance, if k ( a 1 , a 1 ) = 0, the functions k ( s , a 1 ), k ( a 1 , t ) are identically zero.