Numerical Algorithms

, Volume 5, Issue 8, pp 419–424 | Cite as

A unifying approach to the regularization of Fourier polynomials

  • D. de Falco
  • M. Frontini
  • L. Gotusso
Smoothing and Regularization

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 theL2 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
$$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$
(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 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).

Keywords

Fourier Unify Approach Optimal Polynomial Numerical Choice Fourier Polynomial 

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References

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Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • D. de Falco
    • 1
  • M. Frontini
    • 1
  • L. Gotusso
    • 1
  1. 1.Politecnico di MilanoMilanoItaly

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