Estimates for the SVD of the Truncated Fourier Transform on L2(cosh(b.)) and Stable Analytic Continuation

Abstract

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 (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 L 2 (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.

Replaces

Eric Gautier, and Christophe Gaillac, “Estimates for the SVD of the Truncated Fourier Transform on L2(cosh(b.)) and Stable Analytic Continuation”, TSE Working Paper, n. 19-1013, May 2019.

Reference

Christophe Gaillac, and Eric Gautier, “Estimates for the SVD of the Truncated Fourier Transform on L2(cosh(b.)) and Stable Analytic Continuation”, Journal of Fourier Analysis and Applications, vol. 27, n. 72, August 2021.