Fourier frequency adaptive regularization for smoothing data
The authors deal with the problem of smoothing data by a transform in the Fourier domain. For this purpose they consider a regularization problem which generalizes the similar ones analysed by \textit{U. Amato} and \textit{I. De Feis} [J. Comput. Appl. Math. 87, No. 2, 261-284 (1997; Zbl 0893.65073)] in the case of uncorrelated noise, and by \textit{U. Amato} and \textit{I. De Feis} [Appl. Numer. Math., submitted for publication] in the case of correlated noise. The authors prove that under some assumptions the sequence of solutions of the regularization problem converges in the average to the solution of the smoothing data problem and they show that the generalized cross validation (GCV)-like criterion of \textit{U. Amato} and \textit{I. De Feis} is able to estimate, asymptotically, the optimal regularization parameter efficiently in the average. This leads, by minimizing the \(L_2\) risk, to a new adaptive regularization problem, where each Fourier coefficient is smoothed independently and depends on both the noise and the unknown function. Some numerical calculations are done to compare the new adaptive regularization with the GCV regularization of \textit{U. Amato} and \textit{I. De Feis} [Appl. Numer. Math., submitted for publication], with the wavelet regularization of \textit{A. Antoniadis} [ Scand. J. Stat. 23, No. 3, 313-330 (1996; Zbl 0861.62028)] and of \textit{U. Amato} and \textit{D.T. Vuza} [Rev. Roum. Math. Pures Appl. 42, No. 7-8, 481-493 (1997; Zbl 0926.62023)], and with the optimal adaptive wavelet regularization by \textit{U. Amato, M.R. Occorsio} and \textit{D.T. Vuza} [Rend. Circ. Mat. Palermo, II. Ser. 52, 207-222 (1998; Zbl 0935.42019)].
scientific article; zbMATH DE number 1423307
scientific article