# A unifying approach to the regularization of Fourier polynomials

Smoothing and Regularization

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## Abstract

In a previous paper [4] the following problem was considered:, where ∥·∥ is the. In this paper we consider the more general functional, which reduces to (0.1) for

*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)

*L*^{2}norm,*F*_{ n }^{ (r) }is the*r*th derivative of the Fourier polynomial*F*_{ n }(*x*), and*f(x)*is a given function with Fourier coefficients*c*_{ k }. It was proved that the optimal polynomial has coefficients*c*_{ k }^{*}given by$$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$

(0.2)

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

(0.3)

*σ*_{ 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 weights*σ*_{ r }. This result will be used to propose a motivated numerical choice of the parameter*σ* 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