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

Résumé

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.