FOM: Wiener's theorem

[Stephen G Simpson]

Simpson 20 Apr 1999 19:02:32

  Martin Davis 19 Apr 1999 11:06:04 mentions a beautiful theorem of
  Wiener, the original hard analysis proof by Wiener, and the
  subsequent soft proof by Gelfand using Banach algebra methods.  If I
  remember correctly, this is called Wiener's Tauberian theorem.

Martin Davis 20 Apr 1999 16:43:16

  No this is not called Wiener's Tauberian theorem; it is not even a
  Tauberian theorem.

Darn!  Yes, Davis is right and I was wrong.  Normally I would have
checked this before posting, but this semester I don't have access to
my library.  I have now looked it up in a borrowed copy of Rudin's
``Real and Complex Analysis''.  According to Rudin, it is a key step
in Wiener's proof of the Wiener Tauberian theorem.  Furthermore,
although Rudin cites the original hard analysis proof by Wiener, he
only presents details of Gelfand's soft proof.  He says that this soft
proof was one of the early triumphs of Banach algebra theory.

Martin Davis 20 Apr 1999 16:49:46

  I believe there is a ... procedure that would directly (without
  invoking Shoenfield) cut Gelfand's proof down by replacing the
  general theorems using AC (ideal can be extended to maximal ideal &
  Hahn-Banach) by the particular cases actually used, where (I
  conjecture) it can all be made explicit.

This is undoubtedly correct.  The separable case is sufficient.  The
Hahn-Banach theorem for separable Banach spaces is provable in WKL_0
(Brown-Simpson).  I haven't looked at maximal ideals in separable
Banach algebras, but it ought to go through in ACA_0 at worst.  The
reverse mathematics of this could be interesting.  This is one of the
reverse mathematics problems that I keep asking my students to look at
but they keep ignoring ....

-- Steve