Problem 2.20 This problem is designed to guide you through a "proof " of Plan­ cherel' s theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to   infinity.

(a)     Dirichlet's  theorem  says  that  ''any" function  f  (x )  on  the  interval  [-a, +a]

can be expanded  as a Fourier  series:

 

00

f  (x )  = I )a11  sin(nrrx /a ) + b11 cos( nrrx / a) ].

11=0

 

Show that this can be written  equivalently   as

 

00

f (x)  =             c,,eimr x /a .

11=-00

 

What  is  c11 ,  in  terms  of  a11  and  b11 ?

(b)      Show (by appropriate modification of Fourier's trick) that

C11   = _1  f_ +a f (x)e-i mr x /a dx .

2a    -a

(c)     )  Eliminate n and c11  in favor of the new variables k = (nrr / a) and F ( k )  =

,J21ii ac11 •   Show that  (a) and  (b) now become

 

1      00                     . .

r,c    L.,

V .t..JL   11=-00

 

 

F ( k ) =

 

-1    f_ +a f (x )e-,.b .dx ,

v'Zrr   -(I

 

 

where  tik is the increment in k  from one n  to the   next.

 

 

(d)   Take  the  limit  a  ---+   oo  to  obtain  Plancherel's  theorem.  Comment:  In  view of  their  quite  different  origins,  it  is  surprising  (and  delightful )  that  the  two formulas-one  for  F ( k )   in  terms  of  f (x),  the  other  for  f(x)  in  terms  of F(k )-have  such  a similar structure  in  the  limit  a ---+  oo.