[ 111 1 



IV. Slurm-Liouville Series of Normal Functions in the Theory of Integral 



!' < I nations. 



By J. MERCER, M.A. (Cantab.), D.Sc. (Liverpool), Fellow of Trinity 



College, Cambridge. 



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



Received January 20, in revised form December 16, 1909, Read March 3, 1910. 



Introduction. 



ONE of the most important branches of the theory of integral equations is connected 

 with the problem of representing a function as a series of normal functions, 

 HILBERT* and SCHMIDT,! who made the earliest contributions, have been able to 

 obtain sufficient conditions under which an assigned function may be expanded in 

 terms of a system of normal functions belonging to a symmetric characteristic 

 function (kern). These conditions are narrow in respect to the nature of the function 

 which may be expanded, but they have the advantage of applying to very general 

 systems of normal functions. They apply, in particular, to the expansion of a function 

 in both the sine and cosine series of FOURIER. It is in the light of our knowledge of 

 the properties of the latter series that the narrowness above referred to becomes 

 evident. In point of fact, the Hilbert-Schmidt theory is only applicable to FOURIER'S 

 series corresponding to a function which has a continuous second differential coefficient 

 in (0, TT), and which furthermore satisfies certain boundary conditions at the end 

 points of the interval. For example, in the case of the sine series, the function must 

 vanish at both end points. It would appear, therefore, that the wide generality as 

 to the system of normal functions is obtained at the cost of the generality of the 

 function which it is desired to represent. 



Later memoirs by KNESER and HoBSOxJ have made it abundantly clear that, by 

 restricting the nature of the system of normal functions, results may be obtained in 

 regard to the representation of very much wider classes of functions than were 



* 'Gott. Nachr.' (1904), pp. 71-78. 

 t 'Math. Ann.,' vol. 63, pp. 451-453. 



J See also A. C. Dixox, ' Proc. Loud. Math. Soc.,' series 2, vol. 3, pp. 83-103. 

 VOL. CCXI. A 474. 8.5.11 



