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 subspaceiff 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.