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We tackle the inverse problem of reconstructing an unknown finite measure$\mu$ from a noisy observation of a generalized moment of $\mu$ defined as theintegral of a continuous and bounded operator $\Phi$ with respect to $\mu$.When only a quadratic approximation $\Phi_m$ of the operator is known, weintroduce the $L^2$ approximate maximum entropy solution as a minimizer of aconvex functional subject to a sequence of convex constraints. Under severalassumptions on the convex functional, the convergence of the approximatesolution is established and rates of convergence are provided.

Regularization with Approximated $L^2$ Maximum Entropy Method