Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions

We give new formulas for finding a compactly supported function v on Rd, d > 1, from its Fourier transform Fv given within the ball Br. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF’s). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.

[1]  A. Bonami,et al.  Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions , 2015, 1509.02651.

[2]  R. Gerchberg.  Super-resolution through Error Energy Reduction , 1974 .

[3]  P. Maréchal,et al.  A variational approach to the inversion of truncated Fourier operators , 2009 .

[4]  A. Lannes,et al.  Stabilized Reconstruction in Signal and Image Processing , 1987 .

[5]  D. Slepian.  Some comments on Fourier analysis, uncertainty and modeling , 1983 .

[6]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[7]  Vladimir Rokhlin,et al.  Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit , 2007 .

[8]  T. Hohage,et al.  Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems , 2015, 1512.06586.

[9]  Gregory Beylkin,et al.  Nonlinear inversion of a band-limited Fourier transform , 2009 .

[10]  Roman G. Novikov,et al.  New Global Stability Estimates for Monochromatic Inverse Acoustic Scattering , 2013, SIAM J. Math. Anal..

[11]  R. Novikov,et al.  Hölder-logarithmic stability in Fourier synthesis , 2020, Inverse Problems.

[12]  Emmanuel J. Cand.  Towards a Mathematical Theory of Super-Resolution , 2012 .

[13]  Abderrazek Karoui,et al.  Spectral Decay of Time and Frequency Limiting Operator , 2015 .

[14]  Li-Lian Wang,et al.  Analysis of spectral approximations using prolate spheroidal wave functions , 2009, Math. Comput..

[15]  A. Papoulis.  A new algorithm in spectral analysis and band-limited extrapolation. , 1975 .

[16]  Yoel Shkolnisky,et al.  Approximation of bandlimited functions , 2006 .

[17]  Thorsten Hohage,et al.  New Stability Estimates for the Inverse Acoustic Inhomogeneous Medium Problem and Applications , 2001, SIAM J. Math. Anal..

[18]  R. Novikov,et al.  Stability estimates for reconstruction from the Fourier transform on the ball , 2020, 2007.12013.