A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis which prevents the general term of the series from converging to zero too slowly. Hardy (1999, p. 46) states that "a 'Tauberian' theorem may be defined as a corrected form of the false converse of an 'Abelian theorem.' "
Wiener's Tauberian theorem states that if 
, then the translates
of 
span a dense subspace iff the
Fourier transform is nonzero everywhere. This
theorem is analogous with the theorem that if 
(for a Banach algebra
with a unit), then 
spans the whole space if and only if the Gelfand
transform is nonzero everywhere.
