Estimates for the SVD of the truncated Fourier transform on L2(exp(b|$\times$|)) and stable analytic continuation

TL;DR

This paper analyzes the singular value decomposition of the truncated Fourier transform on a weighted L^2 space, providing bounds, convergence rates for analytic continuation, and a numerical method for computation.

Contribution

It extends the analysis of the truncated Fourier transform to larger weighted spaces, offering new bounds, convergence results, and a numerical approach for stable analytic continuation.

Findings

Nonasymptotic bounds on singular values with similar behavior in parameters

Rates of convergence for stable analytic continuation

A numerical method for SVD computation and application

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(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.