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Niveau: Supérieur, Doctorat, Bac+8

How well does the Hermite–Pade approximation smooth the Gibbs phenomenon ? Bernhard Beckermann, Valeriy Kalyagin, Ana C. Matos and Franck Wielonsky January 12, 2010 Abstract In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function f , defined on [?π, π], discontinuous at 0, Driscoll and Fornberg considered so-called singular Fourier-Pade approximants constructed from the Hermite-Pade approximants of the system of functions (1, g1(z), g2(z)), where g1(z) = log(1 ? z) and g2(z) is analytic, such that Re (g2(eit)) = f(t). Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Pade approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we ad- dress the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Pade approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Pade approximants are more efficient than the more elementary Pade approximants, par- ticularly around the discontinuity of the goal function f .

How well does the Hermite–Pade approximation smooth the Gibbs phenomenon ? Bernhard Beckermann, Valeriy Kalyagin, Ana C. Matos and Franck Wielonsky January 12, 2010 Abstract In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function f , defined on [?π, π], discontinuous at 0, Driscoll and Fornberg considered so-called singular Fourier-Pade approximants constructed from the Hermite-Pade approximants of the system of functions (1, g1(z), g2(z)), where g1(z) = log(1 ? z) and g2(z) is analytic, such that Re (g2(eit)) = f(t). Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Pade approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we ad- dress the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Pade approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Pade approximants are more efficient than the more elementary Pade approximants, par- ticularly around the discontinuity of the goal function f .

- lebesgue measure
- goal function
- hermite-pade approximants
- orthogonal polynomials
- very convincing
- convincing numerical
- function g1
- numerical experiments

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Published by | mijec |

Reads | 30 |

Language | English |

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