Convergence for the regularized inversion of Fourier series (Q1379697)

The authors discuss the problem of smoothing data by means of a transform in the Fourier domain and regularization. They prove the convergence of the regularization method when the noise is finite and demonstrate the efficiency of the generalized cross validation criterion for choosing the regularization parameter to be asymptotically optimal. Among others, they show that the error of the regularized solution remains bounded, whereas the error of the non-regularized solution becomes unbounded. Numerical examples are provided to illustrate the rate of convergence and the efficiency of the method proposed.