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Вие сте тук: Начало Колоквиум на ФМИ Резюме на доклад на тема "Stable reconstruction from Fourier samples", професор Шадрин

Резюме на доклад на тема "Stable reconstruction from Fourier samples", професор Шадрин

For an analytic and periodic function $f$, the $m$-th partial sums of its Fourier series converge exponentially fast in $m$, but such rapid
convergence is destroyed once periodicity is no longer present (because of the Gibbs phenomenon at the end-points).

We can restore higher-order convergence, e.g., by reprojecting the slowly convergent Fourier series onto a suitable basis of  orthogonal algebraic polynomials, however, all exponentially  convergent methods appear to suffer from some sort of ill-conditioning,  whereas methods that recover $f$ in a stable manner have a much slower approximation rate.

We give to these observations a definite  explanation in terms of the following  fundamental stability barrier: the best possible  convergence rate for a stable reconstruction from the first $m$ Fourier coefficients is root-exponential in $m$.

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