1272 



z=z iibi( -os (2.T/".r — è"/^)* 



Therefore 



1 



J 



K{.r,i/) cos (2jrki^ - ^((k)d!j = ^,(bk \ ak)co8{2jtkx — U(k). 



o 



lil tlie same way we may prove sin {^jtLc — ^uk) to be a oharac- 

 teristic fiiiictioii (that J is such a fiiuction appears at once). Of 

 course some of the functions (4) may be absent from the set of 

 characteristic functions of A'(.i%,v), viz. in the case the corresponding 

 value of lU -\- l>o ^i" "A -\- ^fc or hi, — Ok be zero. 



^ 3. /\{>',i/) be a kernel of the form 



/C(.r,y)=/(.r h.y) + ./^(^•— V) • • (5) 



f{z) being detined in (0,2), 7(2) in (— i, + i). 



We suppose again (/{ — z) = if{z), so that K{.v,y) becomes symme- 

 trical; we further suppose as before /(^) and (f{z) to be continuous. 

 But iu)w we make an assumption different from (3), viz. 



fiz+l) = —fU) . 7(^ — 1) = - (fi^) .... (6) 

 for O^z^i. 

 The functions 



i^2~cos\{2k—i).T.z — ^(ik\ . ^2sin\{2/c—\yiz--^iik\ . . (7) 

 for /•=: 1,2, . . . . form a system of normalized orthogonal functions 

 foi- all values of j:?/,. Now 



(I .'■ 



If we divide the integral into three parts, one integral from to 1 , 

 another from 1 to .r and a third from to .f (the latter with the 

 negative sign) and make in the second integral the substitution 

 ,; = 1 -j- ?i, that integral becomes 



— if{l+^,)cos\{2k-\):t^,-{2k -i)nx—^;ik\dé, 



so that it cancels the third integral. We thus have 

 1 1 



Cf(.v + r,)cos\{2k— 1 )jr.>/ - ^Jk\dy = j f(^)cos\{2k - 1 ).Tr| - ( 2A- - 1 ).-r.»; - ^^h\ d^ 



X 



= COS \{2k — \)jr.v + hjk\ I f{S,)cos{2k - l)jt^d^ -f 



