%0 Journal Article %T Optimal bounds for inverse problems with Jacobi-type eigenfunctions %+ Laboratoire d'Analyse, Topologie, Probabilités (LATP) %A Willer, Thomas %< avec comité de lecture %@ 1017-0405 %J Statistica Sinica %I Taipei : Institute of Statistical Science, Academia Sinica %V 19 %N 2 %P 785-800 %8 2009 %D 2009 %K statistical inverse problems %K minimax estimation %K second generation wavelets %Z Mathematics [math]/Statistics [math.ST] %Z Statistics [stat]/Statistics Theory [stat.TH]Journal articles %X We consider inverse problems where one wishes to recover an unknown function from the observation of a transformation of it by a linear operator, corrupted by an additive white noise perturbation. We assume that the operator admits a singular value decomposition where the eigenvalues decay in a polynomial way, and where Jacobi polynomials appear as eigenfunctions. This includes, as an application, the well known Wicksell's problem. We determine the asymptotic rate of the minimax risk for this model in a wide framework, considering (Lp) losses (1 < p < \infty), and Besov-like regularity spaces. We draw a comparison with the minimax rates of the deconvolution problem, which appears as a critical case of the Jacobi-type rates. We also establish some new results on the needlets introduced by Petrushev and Xu (2005) which appear as essential tools in this setting. %G English %2 https://hal.science/hal-00133830v2/document %2 https://hal.science/hal-00133830v2/file/newrates.pdf %L hal-00133830 %U https://hal.science/hal-00133830 %~ LATP %~ CNRS %~ UNIV-AMU %~ INSMI %~ I2M