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Dual and Approximately Dual Hilbert–Schmidt Frames in Hilbert Spaces

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Abstract

In this paper, we introduce the notion of approximately dual Hilbert–Schmidt frames. We establish the links between approximately dual Hilbert–Schmidt frames and dual Hilbert–Schmidt frames, and between approximately dual Hilbert–Schmidt frames and approximately dual frames; obtain a procedure to construct approximately dual Hilbert–Schmidt frames; prove that (canonical) dual Hilbert–Schmidt frames inherit the stability of their underlying Hilbert–Schmidt frames. A perturbation theorem of approximately dual Hilbert–Schmidt frames is also presented.

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References

  1. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  2. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  4. Li, Y.-Z., Zhang, W.: Dilation-and-modulation systems on the half real line. J. Inequal. Appl. 186, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)

    MATH  Google Scholar 

  6. Heil, C.: A basis theory primer. Birkhäuser, New York (2010)

    MATH  Google Scholar 

  7. Khosravi, A., Musazadeh, K.: Fusion frames and g-frames. J. Math. Anal. Appl. 342, 1068–1083 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eldar, Y.: Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors. J. Fourier Anal. Appl. 9, 77–96 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Han, D., Leng, J., Huang, T.: Optimal dual frames for communication coding with probabilistic erasures. IEEE Trans. Signal Process. 59, 5380–5389 (2011)

    Article  MathSciNet  Google Scholar 

  10. Han, D., Sun, W.: Reconstruction of signals from frame coefficients with erasures at unknown locations. IEEE Trans. Inform. Theory 60(7), 4013–4025 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feichtinger, H.G., Onchis, D.M., Wiesmeyr, C.: Construction of approximate dual wavelet frames. Adv. Comput. Math. 40, 273–282 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Javanshiri, H.: Some properties of approximately dual frames in Hilbert spaces. Results Math. 70(3–4), 475–485 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Christensen, O., Goh, S.S.: From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa. Appl. Comput. Harmon. Anal. 36, 198–214 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Casazza, P.G.: The art of frame theory. Taiwan. J. Math. 4(2), 129–201 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, X.B., Yang, S.Z., Zhu, Y.C.: Some results about operator perturbation of fusion frames in Hilbert spaces. J. Math. Anal. Appl. 421, 1417–1427 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Christensen, O., Stoeva, D.: \(p\)-frames in separable Banach spaces. Adv. Comput. Math. 18(24), 117–126 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Casazza, P.G., Kutyniok, G.: Frame of subspace. Contemp. Math. 345, 87–114 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sun, W.: G-frames and G-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sadeghi, G., Arefijamaal, A.: von Neumann–Schatten frames in separable Banach spaces. Mediterr. J. Math. 9(3), 525–535 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Arefijamaal, A., Sadeghi, G.: von Neumann–Schatten dual frames and their perturbations. Results Math. 69(3–4), 431–441 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Poria, A.: Some identities and inequalities for Hilbert–Schmidt frames (2016). arXiv preprint arXiv:1602.07912

  22. Ehler, M., Han, B.: Wavelet bi-frames with few generators from multivariate refinable functions. Appl. Comput. Harmon. Anal. 25, 407–414 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Christensen, O., Laugesen, R.S.: Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames. Sampl. Theory Signal Image Process. 9, 77–90 (2011)

    MATH  Google Scholar 

  24. Murphy, G.J.: \(\text{ C }^*\)-Algebra and Operator Theory. Academic Press, San Diego (1990)

    Google Scholar 

  25. Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258, 397–420 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sun, W.C.: Stability of \(g\)-frames. J. Math. Anal. Appl. 326(2), 858–868 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khosravi, A., Azandaryani, M.M.: Approximate duality of \(g\)-frames in Hilbert spaces. Acta. Math. Sci. 34(3), 639–652 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou, 450046, People’s Republic of China

    Wei Zhang

  2. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Wei Zhang

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  1. Wei Zhang
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Correspondence to Wei Zhang.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11271037).

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Zhang, W. Dual and Approximately Dual Hilbert–Schmidt Frames in Hilbert Spaces. Results Math 73, 4 (2018). https://doi.org/10.1007/s00025-018-0793-x

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  • DOI: https://doi.org/10.1007/s00025-018-0793-x

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