[ 415 ] 



XVI. Functions of Positive and Negative Type, and their Connection with the 



Theory of Integral Equations. 



By J. MERCER, B.A., Trinity College, Cambridge. 

 Communicated by Prof. A. R. FORSYTH, Sc.D., LL.D., F.R.S. 



Received December 21, 1908 Read May 13, 1909. 



Introduction, 



THE present memoir is the outcome of an attempt to obtain the conditions under 

 which a given symmetric and continuous function K (.s, t) is definite, in the sense of 

 HILBERT.* At an early stage, however, it was found that the class of definite 

 functions was too restricted to allow the determination of necessary and sufncie?it 

 conditions in terms of the determinants of 10. The discovery that this could be 

 done for functions of positive or negative type, and the fact that almost all the 

 theorems which are true of definite functions are, with slight modification, true of 

 these, led finally to the abandonment of the original plan in favour of a discussion of 

 the properties of functions belonging to the wider classes. 



The first part of the memoir is devoted to the definition of various terms employed, 

 and to the re-statement of the consequences which follow from HILBERT'S theorem. 



In the second part, keeping the theory of quadratic forms in view, the necessary 

 and sufficient conditions, already alluded to, are obtained. These conditions are then 

 applied to obtain certain general properties of functions of positive and negative type. 



Part III. is chiefly devoted to the investigation of a particular class of functions of 

 positive type. In addition, it includes a theorem which shows that, in general, from 

 each function of positive type it is possible to deduce an infinite number of others of 

 that type. 



Lastly, in the fourth part, it is proved that when K (s, t) is of positive or negative 

 type it may be expanded as a series of products of normal functions, and that this 

 series converges both absolutely and uniformly. 



* <G6tt. Nachr.' (1904), Heft I. 

 VOL. CCIX. A 456. 18.10.09 



