Research Article
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Norm retrieval in dynamical sampling form

Year 2024, Volume: 73 Issue: 3, 860 - 874, 27.09.2024

Fatma Bozkurt

https://doi.org/10.31801/cfsuasmas.1408427
https://izlik.org/JA28MT76RD

Abstract

In this article, we study the construction of norm retrievable frames that have a dynamical sampling structure. For a closed subspace $W$ of $R^n$, we show that when the collection of subspaces $\{A^\ell W\}_{i \in I}$ is norm retrievable in $R^n$ for a unitary or Jordan operator $A$, then there always exists a collection of norm retrievable frame vectors that have a dynamical sampling structure in $R^n$.

Keywords

Frame vectors phase retrieval norm retrieval dynamical sampling

References

  • Aceska, R., Aldroubi, A., Davis, J., Petrosyan, A., Dynamical sampling in shift invariant spaces, Commutative and Noncommutative Harmonic Analysis and Applications, 603 (2013), 139–148. https://dx.doi.org/10.1090/conm/603/12047
  • Aceska, R., Tang, S., Furst, V., Dynamical sampling in hybrid shift invariant spaces, Operator Methods in Wavelets, Tilings, and Frames, 626 (2014), 149. https://dx.doi.org/10.1090/conm/626/12500
  • Aceska, R., Kim, Y. H., Scalability of frames generated by dynamical operators, Frontiers in Applied Mathematics and Statistics, 3 (2017). https://doi.org/10.3389/fams.2017.00022
  • Aguilera, A., Cabrelli, C., Carbajal, D., Paternostro, V., Dynamical sampling for shiftpreserving operators, Applied and Computational Harmonic Analysis, 51 (2021), 258–274. https://doi.org/10.1016/j.acha.2020.11.004
  • Aldroubi, A., Davis, J., Krishtal, I., Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off, Journal of Fourier Analysis and Applications, 21(1) (2015), 11–31. https://doi.org/10.1007/s00041-014-9359-9
  • Aldroubi, A., Cabrelli, C., Molter, U., Tang, S., Dynamical sampling, Applied and Computational Harmonic Analysis, 42(3) (2017), 378–401. https://doi.org/10.1016/j.acha.2015.08.014
  • Aldroubi, A., Davis, J., Krishtal, I., Dynamical sampling: Time–space trade-off, Applied and Computational Harmonic Analysis, 34(3) (2013), 495–503. https://doi.org/10.1016/j.acha.2012.09.002
  • Aldroubi, A., Krishtal, I., Tang, S., Phaseless reconstruction from space–time samples, Applied and Computational Harmonic Analysis, 48(1) (2020), 395–414. https://doi.org/10.1016/j.acha.2015.12.004
  • Aldroubi, A., Petrosyan, A., Dynamical sampling and systems from iterative actions of operators, in: Frames and Other Bases in Abstract and Function Spaces, Springer, pp. 15–26, 2017.
  • Bahmanpour, S., Cahill, J., Casazza, P. G., Jasper, J., Woodland, L. M., Phase retrieval and norm retrieval, in: Trends in harmonic analysis and its applications, Vol. 650 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2015), pp. 3–14. https://doi.org/10.1090/conm/650/13047
  • Balan, R., Casazza, P., Edidin, D., On signal reconstruction without phase, Applied and Computational Harmonic Analysis, 20(3) (2006), 345–356. https://doi.org/10.1016/j.acha.2005.07.001
  • Botelho-Andrade, S., Casazza, P. G., Cheng, D., Haas, J., Tran, T. T., Tremain, J. C., Xu, Z., et al., Phase retrieval by hyperplanes, Frames and Harmonic Analysis, 706 (2018), 21–31. http://dx.doi.org/10.1090/conm/706/14217
  • Botelho-Andrade, S., Casazza, P. G., Van Nguyen, H., Tremain, J. C., Phase retrieval versus phaseless reconstruction, Journal of Mathematical Analysis and Applications, 436(1) (2016), 131–137. https://doi.org/10.1016/j.jmaa.2015.11.045
  • Bozkurt, F., Kornelson, K., Norm retrieval from few spatio-temporal samples, Journal of Mathematical Analysis and Applications, 519(2) (2023), 126804. https://doi.org/10.1016/j.jmaa.2022.126804
  • Bozkurt, F., Tensor product of phase retrievable frames, Sinop ¨ Universitesi Fen Bilimleri Dergisi, 7(2) (2022), 142–151. https://doi.org/10.33484/sinopfbd.1211231
  • Cahill, J., Casazza, P. G., Peterson, J.,Woodland, L., Phase retrieval by projections, Houston Journal of Mathematics, 42(2) (2016), 537–558.
  • Casazza, P. G., Ghoreishi, D., Jose, S., Tremain, J. C., Norm retrieval and phase retrieval by projections, Axioms, 6(1) (2017), 6. https://doi.org/10.3390/axioms6010006
  • Casazza, P. G., Woodland, L. M., Phase retrieval by vectors and projections, in: Operator methods in wavelets, tilings, and frames, Vol. 626 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2014), 1–17. https://dx.doi.org/10.1090/conm/626/12501
  • Casazza, P. G., Kutyniok, G., Frames of Subspaces, In Wavelets, Frames and Operator Theory, volume 345 of Contemp. Math., 87–113. Amer. Math. Soc., Providence, RI, 2004.
  • Chen, H., Wang, Z., Gao, K., Hou, Q., Wang, D., Wu, Z., Quantitative phase retrieval in x-ray zernike phase contrast microscopy, Journal of Synchrotron Radiation, 22(4) (2015), 1056–1061. https://doi.org/10.1107/S1600577515007699
  • Christensen, O., An Introduction to Frames and Riesz Bases, 2nd Edition, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, 2016. https://doi.org/10.1007/978-3-319-25613-9
  • Christensen, O., Hasannasab, M., Rashidi, E., Dynamical sampling and frame representations with bounded operators, Journal of Mathematical Analysis and Applications, 463(2) (2018), 634–644. https://doi.org/10.1016/j.jmaa.2018.03.039
  • Duffin, R. J., Schaeffer, A. C., A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72 (1952), 341–366.
  • Edidin, D., Projections and phase retrieval, Applied and Computational Harmonic Analysis, 42(2) (2017), 350–359. https://doi.org/10.1016/j.acha.2015.12.004
  • Kutyniok, G., Okoudjou, K. A., Philipp, F., Tuley, E. K., Scalable frames, Linear Algebra and Its Applications, 438(5) (2012). https://doi.org/10.1016/j.laa.2012.10.046
  • Martin, R. D., Medri, I., Molter, U., Continuous and discrete dynamical sampling, Journal of Mathematical Analysis and Applications, 499(2) (2021), 125060. https://doi.org/10.1016/j.jmaa.2021.125060
  • Hüe, F., Rodenburg, J., Maiden, A., Sweeney, F., Midgley, P., Wave-front phase retrieval in transmission electron microscopy via ptychography, Physical Review B, 82(12) (2010), 121415. https://link.aps.org/doi/10.1103/PhysRevB.82.121415
  • Nakajima, N., Reconstruction of a wave function from the q function using a phase-retrieval method in quantum-state measurements of light, Physical Review A, 59(6) (1999), 4164. https://link.aps.org/doi/10.1103/PhysRevA.59.4164
  • Pinilla, S., Garc´ıa, H., D´ıaz, L., Poveda, J., Arguello, H., Coded aperture design for solving the phase retrieval problem in x-ray crystallography, Journal of Computational and Applied Mathematics, 338 (2018), 111–128. https://doi.org/10.1016/j.cam.2018.02.002
  • Shi, G., Shanechi, M. M., Aarabi, P., On the importance of phase in human speech recognition, IEEE Transactions on Audio, Speech, and Language Processing, 14(5) (2006), 1867–1874. https://doi.org/10.1109/TSA.2005.858512
  • Yu, R. P., Kennedy, S. M., Paganin, D., Jesson, D., Phase retrieval low energy electron microscopy, Micron, 41(3) (2010), 232–238. https://doi.org/10.1016/j.micron.2009.10.010

Year 2024, Volume: 73 Issue: 3, 860 - 874, 27.09.2024

Fatma Bozkurt

https://doi.org/10.31801/cfsuasmas.1408427
https://izlik.org/JA28MT76RD

Abstract

References

  • Aceska, R., Aldroubi, A., Davis, J., Petrosyan, A., Dynamical sampling in shift invariant spaces, Commutative and Noncommutative Harmonic Analysis and Applications, 603 (2013), 139–148. https://dx.doi.org/10.1090/conm/603/12047
  • Aceska, R., Tang, S., Furst, V., Dynamical sampling in hybrid shift invariant spaces, Operator Methods in Wavelets, Tilings, and Frames, 626 (2014), 149. https://dx.doi.org/10.1090/conm/626/12500
  • Aceska, R., Kim, Y. H., Scalability of frames generated by dynamical operators, Frontiers in Applied Mathematics and Statistics, 3 (2017). https://doi.org/10.3389/fams.2017.00022
  • Aguilera, A., Cabrelli, C., Carbajal, D., Paternostro, V., Dynamical sampling for shiftpreserving operators, Applied and Computational Harmonic Analysis, 51 (2021), 258–274. https://doi.org/10.1016/j.acha.2020.11.004
  • Aldroubi, A., Davis, J., Krishtal, I., Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off, Journal of Fourier Analysis and Applications, 21(1) (2015), 11–31. https://doi.org/10.1007/s00041-014-9359-9
  • Aldroubi, A., Cabrelli, C., Molter, U., Tang, S., Dynamical sampling, Applied and Computational Harmonic Analysis, 42(3) (2017), 378–401. https://doi.org/10.1016/j.acha.2015.08.014
  • Aldroubi, A., Davis, J., Krishtal, I., Dynamical sampling: Time–space trade-off, Applied and Computational Harmonic Analysis, 34(3) (2013), 495–503. https://doi.org/10.1016/j.acha.2012.09.002
  • Aldroubi, A., Krishtal, I., Tang, S., Phaseless reconstruction from space–time samples, Applied and Computational Harmonic Analysis, 48(1) (2020), 395–414. https://doi.org/10.1016/j.acha.2015.12.004
  • Aldroubi, A., Petrosyan, A., Dynamical sampling and systems from iterative actions of operators, in: Frames and Other Bases in Abstract and Function Spaces, Springer, pp. 15–26, 2017.
  • Bahmanpour, S., Cahill, J., Casazza, P. G., Jasper, J., Woodland, L. M., Phase retrieval and norm retrieval, in: Trends in harmonic analysis and its applications, Vol. 650 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2015), pp. 3–14. https://doi.org/10.1090/conm/650/13047
  • Balan, R., Casazza, P., Edidin, D., On signal reconstruction without phase, Applied and Computational Harmonic Analysis, 20(3) (2006), 345–356. https://doi.org/10.1016/j.acha.2005.07.001
  • Botelho-Andrade, S., Casazza, P. G., Cheng, D., Haas, J., Tran, T. T., Tremain, J. C., Xu, Z., et al., Phase retrieval by hyperplanes, Frames and Harmonic Analysis, 706 (2018), 21–31. http://dx.doi.org/10.1090/conm/706/14217
  • Botelho-Andrade, S., Casazza, P. G., Van Nguyen, H., Tremain, J. C., Phase retrieval versus phaseless reconstruction, Journal of Mathematical Analysis and Applications, 436(1) (2016), 131–137. https://doi.org/10.1016/j.jmaa.2015.11.045
  • Bozkurt, F., Kornelson, K., Norm retrieval from few spatio-temporal samples, Journal of Mathematical Analysis and Applications, 519(2) (2023), 126804. https://doi.org/10.1016/j.jmaa.2022.126804
  • Bozkurt, F., Tensor product of phase retrievable frames, Sinop ¨ Universitesi Fen Bilimleri Dergisi, 7(2) (2022), 142–151. https://doi.org/10.33484/sinopfbd.1211231
  • Cahill, J., Casazza, P. G., Peterson, J.,Woodland, L., Phase retrieval by projections, Houston Journal of Mathematics, 42(2) (2016), 537–558.
  • Casazza, P. G., Ghoreishi, D., Jose, S., Tremain, J. C., Norm retrieval and phase retrieval by projections, Axioms, 6(1) (2017), 6. https://doi.org/10.3390/axioms6010006
  • Casazza, P. G., Woodland, L. M., Phase retrieval by vectors and projections, in: Operator methods in wavelets, tilings, and frames, Vol. 626 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2014), 1–17. https://dx.doi.org/10.1090/conm/626/12501
  • Casazza, P. G., Kutyniok, G., Frames of Subspaces, In Wavelets, Frames and Operator Theory, volume 345 of Contemp. Math., 87–113. Amer. Math. Soc., Providence, RI, 2004.
  • Chen, H., Wang, Z., Gao, K., Hou, Q., Wang, D., Wu, Z., Quantitative phase retrieval in x-ray zernike phase contrast microscopy, Journal of Synchrotron Radiation, 22(4) (2015), 1056–1061. https://doi.org/10.1107/S1600577515007699
  • Christensen, O., An Introduction to Frames and Riesz Bases, 2nd Edition, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, 2016. https://doi.org/10.1007/978-3-319-25613-9
  • Christensen, O., Hasannasab, M., Rashidi, E., Dynamical sampling and frame representations with bounded operators, Journal of Mathematical Analysis and Applications, 463(2) (2018), 634–644. https://doi.org/10.1016/j.jmaa.2018.03.039
  • Duffin, R. J., Schaeffer, A. C., A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72 (1952), 341–366.
  • Edidin, D., Projections and phase retrieval, Applied and Computational Harmonic Analysis, 42(2) (2017), 350–359. https://doi.org/10.1016/j.acha.2015.12.004
  • Kutyniok, G., Okoudjou, K. A., Philipp, F., Tuley, E. K., Scalable frames, Linear Algebra and Its Applications, 438(5) (2012). https://doi.org/10.1016/j.laa.2012.10.046
  • Martin, R. D., Medri, I., Molter, U., Continuous and discrete dynamical sampling, Journal of Mathematical Analysis and Applications, 499(2) (2021), 125060. https://doi.org/10.1016/j.jmaa.2021.125060
  • Hüe, F., Rodenburg, J., Maiden, A., Sweeney, F., Midgley, P., Wave-front phase retrieval in transmission electron microscopy via ptychography, Physical Review B, 82(12) (2010), 121415. https://link.aps.org/doi/10.1103/PhysRevB.82.121415
  • Nakajima, N., Reconstruction of a wave function from the q function using a phase-retrieval method in quantum-state measurements of light, Physical Review A, 59(6) (1999), 4164. https://link.aps.org/doi/10.1103/PhysRevA.59.4164
  • Pinilla, S., Garc´ıa, H., D´ıaz, L., Poveda, J., Arguello, H., Coded aperture design for solving the phase retrieval problem in x-ray crystallography, Journal of Computational and Applied Mathematics, 338 (2018), 111–128. https://doi.org/10.1016/j.cam.2018.02.002
  • Shi, G., Shanechi, M. M., Aarabi, P., On the importance of phase in human speech recognition, IEEE Transactions on Audio, Speech, and Language Processing, 14(5) (2006), 1867–1874. https://doi.org/10.1109/TSA.2005.858512
  • Yu, R. P., Kennedy, S. M., Paganin, D., Jesson, D., Phase retrieval low energy electron microscopy, Micron, 41(3) (2010), 232–238. https://doi.org/10.1016/j.micron.2009.10.010
There are 31 citations in total.

Details

Primary Language English
Subjects Operator Algebras and Functional Analysis
Journal Section Research Article
Authors

Fatma Bozkurt 0000-0002-6788-0052

Submission Date December 22, 2023
Acceptance Date July 8, 2024
Publication Date September 27, 2024
DOI https://doi.org/10.31801/cfsuasmas.1408427
IZ https://izlik.org/JA28MT76RD
Published in Issue Year 2024 Volume: 73 Issue: 3

Cite

APA Bozkurt, F. (2024). Norm retrieval in dynamical sampling form. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(3), 860-874. https://doi.org/10.31801/cfsuasmas.1408427
AMA 1.Bozkurt F. Norm retrieval in dynamical sampling form. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(3):860-874. doi:10.31801/cfsuasmas.1408427
Chicago Bozkurt, Fatma. 2024. “Norm Retrieval in Dynamical Sampling Form”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 (3): 860-74. https://doi.org/10.31801/cfsuasmas.1408427.
EndNote Bozkurt F (September 1, 2024) Norm retrieval in dynamical sampling form. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 3 860–874.
IEEE [1]F. Bozkurt, “Norm retrieval in dynamical sampling form”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 3, pp. 860–874, Sept. 2024, doi: 10.31801/cfsuasmas.1408427.
ISNAD Bozkurt, Fatma. “Norm Retrieval in Dynamical Sampling Form”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/3 (September 1, 2024): 860-874. https://doi.org/10.31801/cfsuasmas.1408427.
JAMA 1.Bozkurt F. Norm retrieval in dynamical sampling form. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:860–874.
MLA Bozkurt, Fatma. “Norm Retrieval in Dynamical Sampling Form”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 3, Sept. 2024, pp. 860-74, doi:10.31801/cfsuasmas.1408427.
Vancouver 1.Fatma Bozkurt. Norm retrieval in dynamical sampling form. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024 Sep. 1;73(3):860-74. doi:10.31801/cfsuasmas.1408427

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

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