1270 



of (J) fur certain values of ;«. It therefore seems to me to be of some 

 iniportaiice to consider generally the kernels the characteristic func- 

 tions of which satisfy (1). From the examples of § 2 and § 3, con- 

 taining two arbitrary functions, it will a[)pear that very general 

 classes of kernels have solutions of (1) for their characteristic functions. 



§ 2. For ()<.'■< 1 and < // < 1 



K(x,^j)=zF(.x^lj) -\~ <P{.r—j^). ..... (2a) 



will be a symmetrical function of ,r and // under the condition 

 </'(.(■ — y) == </^{i/—.f}. The • function F{z) has to be defined in the 

 interval (0,1), the function '/\'.r) in the interval ( — 1, -|- i). We 

 suppose F{z) and </>(z) to have torO ^. : < i the properties expressed by 

 F(zi\)=F{:) , 0(,_i,= 0(^). .... (3) 

 Supposing the possiitility of expanding F{z) and '/'(:) in uniformly 

 convergent Fourikk series, we have 



00 



00 



'/'(e) = />j + 2 bk cos 2jt/cz, 

 k=\ 



because from '/'( — z}-='/>{z) and ^h{z — 1 ) = </'(2) it follows that 



0(e) = 0(1— ^). 



Thus 



K{x,y) z= a^ - 2 at\i'os {2:xk.v — ^ «a) m^ {2.T^•_v — ^ «t) — 

 k=\ 



— 5?7i (2-TA-.r — ^ a^•) sin {2jt/c>/ — ^ «A;)} 



<x> 



-f /), ^^ Aytloo* (2.T/-.r - i «jfc) ro5 (2.-r/:// ~ ^ «;t) ^- 

 k=l 



-\- sill (2.t/;.c — ^ f^y[) «?« {2:Tky — I ((k)\ 



— (/',H-",) + ^ \(bk^ak) cos (2iiLr - ), ak) cos {2nky - i r^) "h 



^- (bi—ak) sin {2-ik.r — ^ «/,) .s//* (2.7% — ^ (tk)\. 



The functions 



1 , \/¥cos (2jrky - i <a) , K2~sm (2.t% — i uk) . . (4 

 being orihogonal and normalised and the series converging uniformly, 

 it follows by multiplying by one of those functions and integrating 

 with respect to ;/ from to 1, that I he functions (4) will be characteristic 

 functions belonging to the characteristic numbers i {h^-\-a,) , 2/{bk-{-ak) 

 and 2/{hk — dk)- As a further characteristic function of /i^ (a*,//) should 

 have to be continuous and orthogonal to the system (4), it appears 

 that such a function does not exist, the system (4) being complete. 



The supposition of 7^(2) and 0(2) being developable in uniforndy 



