Wiener Tauberian orientation

Wiener's $L^1$ Tauberian theorem states that the linear span of all translates of $f\in L^1({\mathbb{R}}^n)$ is dense in $L^1$ exactly when $\widehat{f}$ has no zeros. This result is recorded, not proved here, and supplies no proof dependency.

Dall'Ara, §3, pp.4–7, proves the convolution-annihilator form: if an $L^1$ family has no common Fourier zero and $g\in L^{\mathrm{\infty }}$ is annihilated by convolution with every member, then $g=0$ almost everywhere (Corollary 3.4). For a singleton family, the usual $L^1$–$L^{\mathrm{\infty }}$ duality and separation of a proper closed subspace translate this into the dense-translate formulation. That equivalence and its functional-analytic assumptions are part of the recorded orientation, not a local proof.

The source's complete route uses its spreading-out lemma, an explicitly convergent Neumann series to solve a convolution equation locally in frequency, and tempered-distribution support. Those later interfaces are not available as proved prerequisites at this location. The no-zero condition here is everywhere nonvanishing; it must not be confused with an $L^2$ density criterion involving almost-everywhere nonvanishing.