A unifying approach to the regularization of Fourier polynomials

Abstract

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:

$$J^* [F_n ,\sigma ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\frac{{\sigma ^r }}{{r!}}} \left\| {F_n^{(r)} } \right\|^2$$

((0.1))

, where ∥·∥ is theL ² norm,F ^(r) _n 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

$$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$

((0.2))

. In this paper we consider the more general functional

$$\hat J[F_n ,\sigma _r ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\sigma _r \left\| {F_n^{(r)} } \right\|^2 }$$

((0.3))

, 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).