0.0.1 Definition and classification of the Wiener algebra
is such that .
0.0.2 Wiener’s Theorem
Theorem (Wiener) - If has no zeros then , that is, has an absolutely convergent Fourier series.
Proof : We want to prove that is invertible in . As is commutative, that is the same as proving that does not belong to any maximal ideal of . Therefore we only need to show that is not in the kernel of any multiplicative linear functional of .
Let be a multiplicative linear functional in . We have that
Since we have that
Since we deduce that
We can conclude that
Therefore we obtain
which is non-zero by definition of .
We conclude that does not belong to the kernel of any multiplicative linear functional .
|Date of creation||2013-03-22 17:22:55|
|Last modified on||2013-03-22 17:22:55|
|Last modified by||asteroid (17536)|