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The problem of solving numerically a Fredholm integral equation of the first kind when it is known that f(x)>or=0, by a regularization method based on minimization is considered. The convergence of solutions of this minimization problem, given conditions on the data and the regularization parameter lambda , is demonstrated to show that the procedure leads to a correct regularization method. A generalized cross validation strategy for the selection of the regularization parameter is introduced. A number of numerical experiments are made and comparisons are made with Tikhonov regularization schemes based on differential operators.

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We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space for "p" ≥ 2. For 1 ≤ "p" > 2, the Bayes Factor is still optimal for (2"s"+2)/(2"s"+1) ≤ "p" > 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..

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