Convergence for the regularized inversion of Fourier series

The problems of smoothing data through a transform in the Fourier domain and of retrieving a function from its Fourier coefficients are analyzed in the present paper. For both of them a solution, based on regularization tools, is known. Aim of the paper is to prove strong results of convergence of the regularized solution and optimality of the Generalized Cross Validation criterion for choosing the regularization parameter.