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In a convolution model, we observe random variables whose distribution is the convolution of some unknown density and some known noise density . We assume that is polynomially smooth. We provide goodness-of-fit testing procedures for the test : = , where the alternative is expressed with respect to ${𝕃}_2$-norm (i.e. has the form ${\psi }_n^{-2}{\parallel f-f_0\parallel }_2^2\ge 𝒞$). Our procedure is adaptive with respect to the unknown smoothness parameter of . Different testing rates ( ...