Hölder-logarithmic stability in the Fourier analysis

Mikhail Isaev (1) , Roman Novikov (2)

1 School of Mathematics [Australia]
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function v on R^d from its Fourier transform Fv given on [−r, r]^d. This estimate relies on a Hölder stable continuation of Fv from [−r, r]^d to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates.

Mots clés: ill-posed inverse problems; Hölder-logarithmic stability; exponential instability; analytic continuation; Chebyshev approximation

Domaines: Mathématiques [math]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]; Physique mathématique [math-ph]

Mikhail Isaev, Roman Novikov. Hölder-logarithmic stability in the Fourier analysis. 2020. ⟨hal-02562474v1⟩