Estimates for the SVD of the Truncated Fourier Transform on $$L^2(\cosh (b|\cdot |))$$ and Stable Analytic Continuation

Abstract

The Fourier transform truncated on \([-c,c]\) has for a long time been analyzed as acting on \(L^2(-1/b,1/b)\) into \(L^2(-1,1)\) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator defined on the larger space \(L^2(\cosh (b|\cdot |))\) on which it remains injective. The main purpose is (1) to provide nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c and (2) to derive nonasymptotic upper bounds on the sup-norm of the right-singular functions. Finally, we propose a numerical method to compute the SVD. These are fundamental results for Gaillac and Gautier (Adaptive estimation in the linear random coefficients model when regressors have limited variation, Bernoulli, arXiv:1905.06584). This paper considers as an illustrative example analytic continuation of nonbandlimited functions when the function is observed with error on an interval.