I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made clearer that such a blur is due to the combining effect of translated images.
I wonder whether this interpretation can be carried on to explain Wiener's Tauberian theorem. I mean the following version
Suppose $\phi\in\mathcal{L}^{\infty}$, and $K\in\mathcal{L}^{1}$ be such that \begin{equation} lim_{x\to\infty} (\phi*K)(x)=a\hat{K}(0),\end{equation}and \begin{equation} \hat{K}(s)\neq 0\end{equation} for all $s$.
Then \begin{equation} lim_{x\to\infty} (\phi*f)(x)=a\hat{f}(0)\end{equation} for all $f\in\mathcal{L}^1$.
So the first equation describe the image in a near-sighted eye, and the last one says something about blurred image in other myopic eyes. But I do not know how to understand the right hand sight of both equations, and how the nonvanishing property relates to eye-sight.
Thanks!