Recent results on the regularization of fourier polynomials

摘要

In a recent paper we presented for two particular cases a unifying approach to the regularization of Fourier polynomials. More precisely, we proved that the regularized polynomials obtained by using the convolution of the given function f(x) with the uniform probability density or with the Gaussian probability density are the same as the ones obtained by minimizing the functional: J∗[Fn∞]=‖f−Fn‖2∑1σrσr‖Fn(r)‖2, where ‖ · ‖ is the L2 norm, F(r)n is the rth derivative of the Fourier polynomial Fn(x), f(x) is a given function with Fourier coefficients ck, and σr are suitable weights. In both cases we have given explicit expressions of the weights σr in the dependence on a scalar parameter τ.In this paper we prove that this unifying approach may be extended to a wide class of convolution kernel. A characterization of this class is also given.