ON THE THEORY OF INTEGRAL EQUATIONS. 301 



is reduced to a canonical form by multiplying it by p(x)k(x,s), where p(x) 



is a positive function and integrating ' between suitable limits a',b'. If 



the new equation is 



H s ) = U\x)p{x)k{x,s)cIv 



a' 



V 



h 

 h(s) = [g{*$4>(t)at. 



Tbe new kernel g(s,t) is a symmetric function of s and t and is such that 



the double integral 



b b 



f U»,t)x(*)x(t)dsdt 



is either positive or zero for any arbitrary continuous function \(t), or 

 in fact for any summable function whose square is also summable. 

 Such a kernel is said to be of positive type? A function g(s,t) which is 

 such that the double integral is always positive is called a definite func- 

 tion (Hilbert) ; this, of course, is an extension of the idea of a definite 

 quadratic form. 



The sum of a definite function and a set of functions of positive type 

 is clearly a definite function ; this fact and the above method of con- 

 structing functions of positive type enable us to form many types of 

 definite functions. 



It has been shown by the author and by J. Mercer that if a(x) is a 

 real continuous function which is never negative and never decreases as 

 x increases from a to b, then the function g{x,t) defined by the equation 



g(x,t) = u(l) a <*<£<& 

 = n (x) a 5 CCr t <b 



is a definite function. From this it follows that if p(.r) y(x) are con 

 tinuous functions which are never negative, and if y(x) increases with x 

 and /i(.r) at the same time decreases, then the function 



h{x,t) = n{t)y(x) x<t 

 = (3(x)y{t) x>t 



is definite. 3 The functions f3(t), >(/) may vanish at the upper and lower 

 limit respectively. 



1 If f(x) is only known for a countable set of values of .r, we must sum instead of 



integrating. If, however, Lebesgue integrals are used tbe distinction is unnecessary. 



- This nomenclature is due to Mercer, Phil. Trans. A., vol. ccis., pp. 115-11 G. 



3 Tbe conditions are satislied for llilbcrt's fund ion 



If *\ x (1 -t) x < t 



h(.x,t) = v ' ■ 



