In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener.[1][2]

Statement

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  • Given a real or complex Borel measure   on the unit circle  , let   be its atomic part (meaning that   and   for  . Then
 

where   is the  -th Fourier coefficient of  .

  • Similarly, given a real or complex Borel measure   on the real line   and called   its atomic part, we have
 

where   is the Fourier transform of  .

Proof

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  • First of all, we observe that if   is a complex measure on the circle then
 

with  . The function   is bounded by   in absolute value and has  , while   for  , which converges to   as  . Hence, by the dominated convergence theorem,

 

We now take   to be the pushforward of   under the inverse map on  , namely   for any Borel set  . This complex measure has Fourier coefficients  . We are going to apply the above to the convolution between   and  , namely we choose  , meaning that   is the pushforward of the measure   (on  ) under the product map  . By Fubini's theorem

 

So, by the identity derived earlier,   By Fubini's theorem again, the right-hand side equals

 
  • The proof of the analogous statement for the real line is identical, except that we use the identity
 

(which follows from Fubini's theorem), where  . We observe that  ,   and   for  , which converges to   as  . So, by dominated convergence, we have the analogous identity

 

Consequences

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  • A real or complex Borel measure   on the circle is diffuse (i.e.  ) if and only if  .
  • A probability measure   on the circle is a Dirac mass if and only if  . (Here, the nontrivial implication follows from the fact that the weights   are positive and satisfy  , which forces   and thus  , so that there must be a single atom with mass  .)

References

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  1. ^ Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle (MathOverflow)
  2. ^ A complex borel measure, whose Fourier transform goes to zero (MathOverflow)