A unifying approach to the regularization of Fourier polynomials

1572-9265

Springer Online Journal Archives 1860-2000

Computer Science

Mathematics

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

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