Estimates for the SVD of the truncated Fourier transform on L2(cosh(b·)) and stable analytic continuation

Christophe Gaillac (1, 2) , Eric Gautier (1, 3)

1 TSE-R - Toulouse School of Economics
2 CREST - Centre de Recherche en Économie et Statistique
3 UT Capitole - Université Toulouse Capitole

The Fourier transform truncated on [−c, c] is usually analyzed when acting on L2(−1/b, 1/b) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator acting on the larger space L2(cosh(b·)) on which it remains injective. We give nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c. The lower bounds are used to obtain rates of convergence for stable analytic continuation of possibly nonbandlimited functions which Fourier transform belongs to L2(cosh(b·)). We also derive bounds on the sup-norm of the singular functions. Finally, we provide a numerical method to compute the SVD and apply it to stable analytic continuation when the function is observed with error on an interval.

Mots clés: Analytic continuation; Nonbandlimited functions; Uniform estimates; Extrapolation; Singular value decomposition; Truncated Fourier transform; Singular Sturm Liouville Equations; Super-resolution

Domaines: Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]; Analyse numérique [math.NA]; Statistiques [math.ST]

Christophe Gaillac, Eric Gautier. Estimates for the SVD of the truncated Fourier transform on L2(cosh(b·)) and stable analytic continuation. 2019. ⟨hal-02130626v1⟩