Phase retrieval in infinite-dimensional Hilbert spaces
HTML articles powered by AMS MathViewer
- by Jameson Cahill, Peter G. Casazza and Ingrid Daubechies HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 3 (2016), 63-76
Abstract:
The main result of this paper states that phase retrieval in infinite-dimensional Hilbert spaces is never uniformly stable, in sharp contrast to the finite-dimensional setting in which phase retrieval is always stable. This leads us to derive stability results for signals depending on how well they are approximated by finite expansions.References
- R. Alaifari and P. Grohs, Phase retrieval in the general setting of continuous frames for Banach spaces, arXiv preprint (2016) arXiv:1604.03163v1.
- R. Balan, Stability of phase retrievable frames, SPIE Optical Engineering + Applications, International Society for Optics and Photonics, 2013; DOI: 10.1117/12.2026135.
- Radu Balan, Pete Casazza, and Dan Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal. 20 (2006), no. 3, 345–356. MR 2224902, DOI 10.1016/j.acha.2005.07.001
- Radu Balan and Yang Wang, Invertibility and robustness of phaseless reconstruction, Appl. Comput. Harmon. Anal. 38 (2015), no. 3, 469–488. MR 3323113, DOI 10.1016/j.acha.2014.07.003
- Afonso S. Bandeira, Jameson Cahill, Dustin G. Mixon, and Aaron A. Nelson, Saving phase: injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal. 37 (2014), no. 1, 106–125. MR 3202304, DOI 10.1016/j.acha.2013.10.002
- Ole Christensen, An introduction to frames and Riesz bases, 2nd ed., Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, [Cham], 2016. MR 3495345, DOI 10.1007/978-3-319-25613-9
- Aldo Conca, Dan Edidin, Milena Hering, and Cynthia Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal. 38 (2015), no. 2, 346–356. MR 3303679, DOI 10.1016/j.acha.2014.06.005
- Stéphane Mallat and Irène Waldspurger, Phase retrieval for the Cauchy wavelet transform, J. Fourier Anal. Appl. 21 (2015), no. 6, 1251–1309. MR 3421917, DOI 10.1007/s00041-015-9403-4
- Volker Pohl, Fanny Yang, and Holger Boche, Phaseless signal recovery in infinite dimensional spaces using structured modulations, J. Fourier Anal. Appl. 20 (2014), no. 6, 1212–1233. MR 3278866, DOI 10.1007/s00041-014-9352-3
- Gaurav Thakur, Reconstruction of bandlimited functions from unsigned samples, J. Fourier Anal. Appl. 17 (2011), no. 4, 720–732. MR 2819174, DOI 10.1007/s00041-010-9144-3
- C. Vinzant, A small frame and a certificate of its injectivity, arXiv preprint (2015) arXiv:1502.04656.
Additional Information
- Jameson Cahill
- Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
- MR Author ID: 972323
- Peter G. Casazza
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 45945
- Ingrid Daubechies
- Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708
- MR Author ID: 54800
- ORCID: 0000-0002-6472-1056
- Received by editor(s): January 27, 2016
- Received by editor(s) in revised form: June 21, 2016
- Published electronically: October 6, 2016
- Additional Notes: The second author was supported by NSF DMS 1609760; NSF ATD 1321779; and ARO W911NF-16-1-0008
The third author was supported by AFOSR grant 00002113-02; ONR grant N00014-11-1-0714-06-7; and NSF grant DMS-1516988 - © Copyright 2016 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 3 (2016), 63-76
- MSC (2010): Primary 46C05; Secondary 94A15
- DOI: https://doi.org/10.1090/btran/12
- MathSciNet review: 3554699