RECENT RESULTS ON THE REGULARIZATION OF FOURIER POLYNOMIALS

In a recent paper we presented for two particular cases a unifying app roach to the regularization of Fourier polynomials. More precisely, we proved that the regularized polynomials obtained by using the convolu tion of the given function f(x) with the uniform probability density o r with the Gaussian probability density are the same as the ones obtai ned by minimizing the functional: GRAPHICS where parallel-to . paralle l-to is the L2 norm, F(n)(r) rth derivative of the Fourier polynomial F(n)(x), f(x) is a given function with Fourier coefficients c(k), and sigma(r) are suitable weights. In both cases we have given explicit ex pressions of the weights sigma(r) in their dependence on a scalar para meter tau. In this paper we prove that this unifying approach may be e xtended to a wide class of convolution kernel. A characterization of t his class is also given.