1271 



eoiivergent Fouuiek series is iiiiiieoessaij, as appears IVom another 

 arrangement of the proof. Suppose F{z) and ^{z) to be continuous 

 functions satisfying onlj the functional equations (3) and the equation 



</>(— ^r) == <h{z). Then 



I F{x 4" y) cos (2 riki/ — 4^(^.) (/// z= I F{i:) cos {2:ik^ — 2jtk.c — l^a/.yi^. 



.,■ 



The integrand havijig the period 1, it is allowed lo integrate from 

 to i instead of from ,c to .i-fl. This gives 

 1 1 



j F(.'i;^y) cos (2riky - ii(k)d!/ = j F{^) cos {2jTk:^—2iik.v—^uk)iri 



Ü 



1 1 



= cos (2rr^.f ] ^tik) i F{§) cos {27Tk'é)di + sm (2jTk.v ^ i^r/,) j F{^)s;H{2ik'^)d'^. 







Now, if 



a, -(- -^ ^A ''ö* {2jTkz — a/c) 

 k=i 



be (he Fourier series of F{2) (no supposition is made on the con- 

 vergence), we have 



1 1 



I F(t) cos (2jfk^)d^ = Uifc ''OS Ok, I F{i) sin {27rk'é)d% = hak sin uk 



(t 



and with this definition of itk and <ik we have 



1 



\F{x-^y)cos(^27iky~{ak)dy^zi 







= ^ayiosakCos{27tkx 4- ^nk) -\- ^a]csinaj^in{2crk.r. -\- i«jt)=: ^akCos{2nkx — ^<f^). 

 Suppose further 



/»„ -f- J" hkCos{2^kz) 

 k=\ 



to be the Fourier series of *I*{z), i.e. 



1 1 



I 0(^) cos {2nki,)dk = Uk . I <r^{i) sin (2-Tki;)d^ =Q. 



II 



Then, fp{z) having the period 1, 

 1 i 



j <r*{x -y) cos (2 iky - huk)dy =i | <I»{i) cos (2.7^§ f 2 //.u- --.i^fA:)*;^ 



