Multiple Fourier series procedures for extraction of nonlinear regressions from noisy data

Three nonparametric procedures for the extraction of nonlinear regressions from noisy data are proposed. The procedures are based on the Dirichlet, Fejer, and de la Vallee Poussin multiple kernels. Convergence properties are investigated. In particular, it is shown that the algorithms are convergent in the mean-integrated-square-error sense. The appropriate theorem establishes a relation between the order of kernels and the number of observations. Special attention is focused on the two-dimensional case. It is proved that the procedures attain the optimal rate of convergence, which cannot be exceeded by any other nonparametric algorithm