Fourier frequency adaptive regularization for smoothing data

The problem of smoothing data through a transform in the Fourier domain is analyzed. It is well known that this problem has a very easy solution and optimal convergence properties; moreover, the GCV criterion is able to give an estimate of the regularization parameter that is asymptotically optimal in the average. The presence of just one regularization parameter in the problem means that all Fourier coefficients are smoothed with the same law, regardless of the function. Here we introduce a frequency adaptive regularization method where a regularization parameter is introduced for each coefficient, able to smooth different frequencies taking into account both function and noise. We give convergence results for the method; moreover an ideal choice of the regularization parameters is provided basing on the minimization of the L2 risk. Numerical experiments are worked out on some significant test functions in order to show performance of the method. Comparison with results achievable with the wavelet regularization and the wavelet adaptive regularization methods is finally performed.