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Estimates for the SVD of the truncated Fourier transform on L2(exp(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(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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https://hal.archives-ouvertes.fr/hal-02130626
Contributor : Eric Gautier <>
Submitted on : Tuesday, April 20, 2021 - 10:07:37 AM
Last modification on : Wednesday, June 9, 2021 - 10:00:20 AM

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  • HAL Id : hal-02130626, version 6
  • ARXIV : 1905.11338

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Christophe Gaillac, Eric Gautier. Estimates for the SVD of the truncated Fourier transform on L2(exp(b|·|)) and stable analytic continuation. 2021. ⟨hal-02130626v6⟩

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