Tensor Product of Phase Retrievable Frames

Yıl 2022, Cilt: 7 Sayı: 2, 142 - 151, 30.12.2022

Fatma Bozkurt

https://doi.org/10.33484/sinopfbd.1211231

Cited By: 1

Öz

Frame vectors in the tensor product of Hilbert spaces that accomplish phase retrieval can be characterized. In this article, we determine the conditions under which the tensor product of vectors may do phase retrieval. Given that tensor product of two frames always implies a frame in the tensor product of Hilbert spaces, we particularly concentrate on finding conditions for phase retrieval in the tensor product of Hilbert spaces.

Anahtar Kelimeler

Frame vectors, Tensor product, Phase Retrieval

Fazları Geri Alınabilen Frame Vektörlerinin Tensör Çarpımı

Öz

Hilbert uzaylarının tensör çarpımındaki faz geri dönüşünü gerçekleştiren frame vektörleri karakterize edilmektedir. Bu
makalede, vektörlerin tensör çarpımının hangi koşullar altında faz
geri getirme yapabileceği belirlenmektedir. Iki frame setinin tensör çarpımının her zaman Hilbert uzaylarının tensör çarpımında bir frame ima ettiği göz önüne alındığında, bu çalışmada özellikle Hilbert uzaylarının tensör çarpımında frame vektörlerinin faz geri dönüşü için gerekli koşullar belirlenmiştir.

Anahtar Kelimeler

Frame vektörleri, Tensör çarpımı, Faz geri alma

Kaynakça

• Duffin, R. J., & Schaeffer, A. C. (1952). A class of nonharmonic fourier series. Transactions of the American Mathematical Society, 72(2), 341–366. https://doi.org/10.2307/1990760
• Gabor, D. (1946). Theory of communication. part 1: The analysis of information. Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, 93(26), 429–441. https://doi.org/10.1049/ji-3-2.1946.0074
• Balan, R., Casazza, P., & Edidin, D. (2006). On signal reconstruction without phase. Applied and Computational Harmonic Analysis, 20(3), 345–356. https://doi.org/10.1016/j.acha.2005.07.001
• Chen, H., Wang, Z., Gao, K., Hou, Q., Wang, D., & Wu, Z. (2015). Quantitative phase retrieval in x-ray zernike phase contrast microscopy. Journal of Synchrotron Radiation, 22(4), 1056–1061. https://doi.org/10.1107/S1600577515007699
• Pinilla, S., García, H., Díaz, L., Poveda, J., & Arguello, H. (2018). Coded aperture design for solving the phase retrieval problem in x-ray crystallography. Journal of Computational and Applied Mathematics, 338, 111-128. https://doi.org/10.1016/j.cam.2018.02.002
• Shi, G., Shanechi, M. M., & Aarabi, P. (2006). On the importance of phase in human speech recognition. IEEE Transactions on Audio, Speech, and Language Processing, 14(5), 1867-1874. https://doi.org/ 10.1109/tsa.2005.858512
• Hüe, F., Rodenburg, J., Maiden, A., Sweeney, F., & Midgley, P. (2010). Wave-front phase retrieval in transmission electron microscopy via ptychography. Physical Review B, 82(12), 121415. https://doi.org/10.1103/PhysRevB.82.121415
• Yu, R. P., Kennedy, S. M., Paganin, D., & Jesson, D. (2010) Phase retrieval low energy electron microscopy. Micron, 41(3), 232-238. https://doi.org/10.1016/j.micron.2009.10.010
• Folland, G. B. (1994). A Course in Abstract Harmonic Analysis. CRC Press.
• Ringrose, J. R., & Kadison, R. V. (1983). Fundamentals of the Theory of Operator Algebras. Academic Press.
• Reddy, G. U., Reddy, N. G., & Reddy, B. K. (2009). Frame operator and hilbert-schmidt operator in tensor product of hilbert spaces. Journal of Dynamical Systems and Geometric Theories, 7(1), 61-70. https://doi.org/10.1080/1726037X.2009.10698563
• Khosravi, A., & Asgari, M. S. (2012). Frames and bases in tensor product of hilbert spaces. arXiv Preprint arXiv:1204.0096. https://doi.org/10.48550/arXiv.1204.0096
• Wang, Y. H., & Li, Y. Z. (2019). Tensor product dual frames. Journal of Inequalities and Applications, 2019(1), 1-17. https://doi.org/10.1186/s13660-019-2034-6
• Zakeri, S., & Ahmadi, A. (2020). Scalable frames in tensor product of hilbert spaces. International Journal of Nonlinear Analysis and Applications, 11(2), 149-159. https://dx.doi.org/10.22075/ijnaa.2019.17608.1953
• Bahmanpour, S., Cahill, J., Casazza, P. G., Jasper, J., & Woodland, L. M. (2014). Phase retrieval and norm retrieval. arXiv Preprint arXiv:1409.8266. https://doi.org/10.48550/arXiv.1409.8266
• Cahill, J., Casazza, P. G., Peterson, J., & Woodland, L. (2013). Phase retrieval by projections. arXiv Preprint arXiv:1305.6226. https://doi.org/10.48550/arXiv.1305.6226
• Casazza, P. G., & Kutyniok, G. (2012). Finite frames: Theory and Applications. Springer.
• Casazza, P. G., & Woodland, L. M. (2014). Phase retrieval by vectors and projections. Operator Methods in Wavelets, Tilings, and Frames, 626, 1-17. https://doi.org/10.1090/conm/626/12501
• Christensen, O. (2003). An Introduction to Frames and Riesz Bases (Vol. 7). Boston: Birkhäuser.