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A note on the phase retrieval of holomorphic functions

Part of: Entire and meromorphic functions, and related topics Spaces and algebras of analytic functions Communication, information

Published online by Cambridge University Press:  08 October 2020

Rolando Perez III
Rolando Perez III*
Affiliation:
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400Talence, France and Institute of Mathematics, University of the Philippines Diliman, 1101 Quezon City, Philippines
*
e-mail:roperez@u-bordeaux.fr
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Abstract

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Keywords

Phase retrievalholomorphic functions

MSC classification

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Secondary: 30H10: Hardy spaces 30H15: Nevanlinna class and Smirnov class 94A12: Signal theory (characterization, reconstruction, filtering, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 779 - 786
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author was supported by the CHED-PhilFrance scholarship from Campus France and the Commission of Higher Education (CHED), Philippines.

References

Akutowicz, E. J., On the determination of the phase of a Fourier integral. I . Trans. Amer. Math. Soc. 83(1956), 179–192. http://dx.doi.org/10.2307/1992910 Google Scholar
Akutowicz, E. J., On the determination of the phase of a Fourier integral. II . Proc. Amer. Math. Soc. 8(1957), 234–238. http://dx.doi.org/10.2307/2033718 Google Scholar
Boche, H., Li, N., and Pohl, V., Phase retrieval in spaces of analytic functions on the unit disk . In: 2017 International Conference on Sampling Theory and Applications (SampTA), Tallin, 2017, pp. 336–340. http://dx.doi.org/10.1109/SAMPTA.2017.8024411 Google Scholar
Bodmann, B. and Hammen, N., Stable phase retrieval with low redundancy frames . Adv. Comput. Math. 41(2015), 317–331. http://dx.doi.org/10.1007/s10444-014-9359-y CrossRefGoogle Scholar
Duren, P., The theory of ${H}^p$ spaces. Academic Press, New York, 1970.Google Scholar
Grohs, P., Koppensteiner, S., and Rathmair, M., Phase retrieval: uniqueness and stability . SIAM Rev. 62(2020), 301–350. http://dx.doi.org/10.1137/19M1256865 CrossRefGoogle Scholar
Grohs, P. and Rathmair, M., Stable Gabor phase retrieval and spectral clustering . Comm. Pure Appl. Math. 72(2019), 981–1043. http://dx.doi.org/10.1002/cpa.21799 CrossRefGoogle Scholar
Hofstetter, E., Construction of time-limited functions with specified autocorrelation functions . IEEE Trans. Inform. Theory 10(1964), 119–126. http://dx.doi.org/10.1109/TIT.1964.1053648 CrossRefGoogle Scholar
Jaming, P., Uniqueness results in an extension of Pauli’s phase retrieval problem . Appl. Comput. Harmon. Anal. 37(2014), 413–441. http://dx.doi.org/10.1016/j.acha.2014.01.003 CrossRefGoogle Scholar
Jaming, P., Kellay, K., and Perez, R. III, Phase retrieval for wide band signals . J. Fourier Anal. Appl. 26(2020), Paper No. 54. http://dx.doi.org/10.1007/s00041-020-09767-1 CrossRefGoogle Scholar
Klibanov, M., Sacks, P., and Tikhonravov, A., The phase retrieval problem . Inverse Prob. 11(1995), 1–28.CrossRefGoogle Scholar
McDonald, J., Phase retrieval and magnitude retrieval of entire functions . J. Fourier Anal. Appl. 10(2004), 259–267. http://dx.doi.org/10.1007/s00041-004-0973-9 CrossRefGoogle Scholar
Nikolski, N., Hardy spaces . Cambridge Studies in Advanced Mathematics , 179, Cambridge University Press, Cambridge, UK, 2019.Google Scholar
Waldspurger, I. and Mallat, S., Phase retrieval for the Cauchy wavelet transform . J. Fourier Anal. Appl. 21(2015), 1251–1309. http://dx.doi.org/10.1007/s00041-015-9403-4 Google Scholar
Walther, A., The question of phase retrieval in optics . Optica Acta 10(1963), 41–49. http://dx.doi.org/10.1080/713817747 CrossRefGoogle Scholar