ON THE THEORY OF INTEGRAL EQUATIONS. 387 



now ground has been broken by Dixon, 1 Kneser, 2 Stekloff, 3 Hilbcrt, 4 

 Schmidt, 5 Mason/'' Hobson, 7 and Filon.* 



In Hilbert's expansion theorem a function which can be expressed m 

 the form 



6 



/(a,) = \G{x,t)t{t)dt (1) 



where G(x,t) is a symmetrical kernel and f(t) a continuous function, can 

 be expanded in an absolutely and uniformly convergent series of the form 



CO 



f(x) = 2 a,^ H {x) 



where the functions f >t (x) are a complete system of orthogonal functions 

 derived from the kernel G(x,t). By taking G(x,t) as the Green's function 

 of a linear differential equation the expansion theorem for series of 

 solutions of the differential equation is obtained, and the condition (1) 

 may be usually replaced by the condition that f(x) possesses a continuous 

 second derivative and satisfies the boundary conditions. 



By means of this theorem all the known expansion theorems— e.g., in 

 series of Bessel, Sturm, Legendre and Lame's functions and also those 

 discussed by Poincare— the normal functions of the differential equation 



8% dH 9% XM = 



dx* df dz 1 



investigated by Le Roy, Stekloff, Zaremba, Korn, and others, are made to 

 depend upon the properties of an integral equation of the type 



f(s) = «/,(,) - \y(s,t)f(t)dt 



where O(s,0] 2 f?s^ 



b b 



[ j[«(*.0] 5 



has a finite value. 



A simplified proof of Hilbert's theorem was given by Schmidt and a 

 restriction was removed. Schmidt also extended the theorem by intro- 

 ducing two systems of orthogonal functions </>„(s), xf/ n (t) connected with an 

 unsymmetrical kernel. These functions are defined by the equations 



a 

 b 



&,(<) = (V»(«) <s,t)ds 



1 Proc. London Math. Soc, ser. 2, vol. iii. 



2 Math. Ann., vols, lviii., lx., lxiii. 



3 Annate* de Tonlouse, 2nd ser., t. 3 ; Comptes rendu*, April 8, 1907 



4 Gottinger Nachriehten, 1901. 



5 Dissertation Gbttingen, 1905; Math. Ann., 1907, vol. lxiii. 



6 Trans. Amcr. Math. Soc, 1907, vol. viii., p. 427. 



7 Proc. London Math. Soc., ser. 2, vol. vi., p. 319 (1908). 



8 Jl'id., ser. 2, vol.iv., p. 396 (1906). 



