Let $K\in L^1(\mathbb R)$ and let $f\in L^\infty(\mathbb R)$ be two functions on $\mathbb R$. The convolution $$(K\ast f)(x)=\int_{ \mathbb R}K(x-y)f(y)dy$$ can be considered as an average of $f$ with weight defined by $K$. Wiener's Tauberian theorem says that under suitable conditions, if $$\lim_{ x\to \infty}(K\ast f)(x)=\lim_{ x\to \infty} (K\ast A)(x)$$ for some constant $A$, then $$\lim_{ x\to \infty}f(x)=A.$$ We prove the following $\ell$-adic analogue of this theorem: Suppose $K,F, G$ are perverse $\ell$-adic sheaves on the affine line $\mathbb A$ over an algebraically closed field of characteristic $p$ ($p\not=\ell$). Under suitable conditions, if $$(K\ast F)|_{ \eta_\infty}\cong (K\ast G)|_{ \eta_\infty},$$ then $$F|_{ \eta_\infty}\cong G|_{ \eta_\infty},$$ where $\eta_\infty$ is the spectrum of the local field of $\mathbb A$ at $\infty$.
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