2019/05/17 by Christophe Gaillac, Gaillac, Christophe, Éric Gautier +1
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1905.11338
openalex publication_date 2019/05/17 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28
The Fourier transform truncated on [-c,c] is usually analyzed when acting on L2(-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 L2(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 L2(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.