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arXiv · 1905.11338

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

Christophe Gaillac·Eric Gautier
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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.

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BibTeXRIS

Christophe Gaillac, Eric Gautier. 2019-05-17. Estimates for the SVD of the truncated Fourier transform on L2(exp(b|$\times$|)) and stable analytic continuation. https://arxiv.org/abs/1905.11338

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