A unifying approach to the regularization of Fourier polynomials

In a previous paper [4] the following problem was considered:find, in the class of Fourier polynomials of degree n, the one which minimizes the functional: (0.1) $$J^* [F_n ,sigma ] = left| {f - F_n } right|^2 + sumlimits_{r = 1}^infty {frac{{sigma ^r }}{{r!}}} left| {F_n^{(r)} } right|^2$$ , where ∥·∥ is theL ² norm,F _n ^(r) is therth derivative of the Fourier polynomialF _n (x), andf(x) is a given function with Fourier coefficientsc _k . It was proved that the optimal polynomial has coefficientsc _k ^* given by (0.2) $$c_k^* = c_k e^{ - sigma k^2 } ; k = 0, pm ,..., pm n$$ . In this paper we consider the more general functional (0.3) $$hat J[F_n ,sigma _r ] = left| {f - F_n } right|^2 + sumlimits_{r = 1}^infty {sigma _r left| {F_n^{(r)} } right|^2 }$$ , which reduces to (0.1) forσ _r =σ ^r /r!. We will prove that the classical sigma-factor method for the regularization of Fourier polynomials may be obtained by minimizing the functional (0.3) for a particular choice of the weightsσ _r . This result will be used to propose a motivated numerical choice of the parameterσ in (0.1).