Estimates for the SVD of the truncated Fourier transform on L2(exp(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 L^2(-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 L^2(exp(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 whose Fourier transform belongs to L^2(exp(b|.|)). We also derive bounds on the sup-norm of the singular functions. Finally, we propose 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: Truncated Fourier transform; Extrapolation; Analytic continuation; Uniform estimates; Singular value decomposition; Nonbandlimited functions; Super-resolution; Singular Sturm Liouville Equations

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(exp(b|·|)) and stable analytic continuation. 2020. ⟨hal-02130626v5⟩