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For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part $$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (by elements of $\mathbb R^n$) can approximate $R(f)$ under $L^1$-norm? We know that if Fourier transform of $f$ is nowhere vanishing then the $L^1$-closure of span of translates of $f$ is $L^1(\mathbb R^n)$, so in particular it contains $R(f)$. But is it a necessary condition?

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