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Generalized framing and Riesz-dual sequences in Banach spaces

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Abstract

The duality principle in frame theory states that for each sequence in a separable Hilbert space, there exists a Riesz-dual sequence dependent only on two orthonormal bases. Then characterize the exact properties of the first sequence in terms of the Riesz-dual sequence, which yield duality relations for the frames. In this paper, we introduce the concept of Riesz-dual (R-dual) sequence with respect to a BK-space in arbitrary Banach spaces. This was referred to as an \(X_d\)-R-dual. We also study the relationship between the R-duals and g-framings. We give conditions under which a pair of R-duals generates a framing. Another important component of this article is an extensive study of the robustness of framing and g-framing systems under small perturbations. These properties of framings and g-framings are completely constructive, relying on a flexible and elementary method for constructing them.

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References

  1. Aldroubi, A., Sun, Q., Tang, W.S.: p-Frames and shift invariant subspaces of \(L^p\). J. Fourier Anal. Appl. 7(1), 1–21 (2001)

    Article  MathSciNet  Google Scholar 

  2. Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)

    Article  MathSciNet  Google Scholar 

  3. Antoine, J.-P., Balazs, P.: Frames and semi-frames, J. Phys. A: Math. Theor. 44, 479501 (2pp) (2011); Corrigendum, 44, 205201 (2011)

  4. Antoine, J.-P., Speckbacher, M., Trapani. C.: Reproducing pairs of measurable functions. Acta Appl. Math. 150(1) 81–101 (2017)

  5. Benedetto, J.J., Frazier, M.W.: Wavelets: mathematics and applications. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  6. Benedetto, J.J., Powell, A.M., Yilmaz, Ö.: Sigma-Delta \((\Sigma \Delta )\) quantization and finite frames. IEEE Trans. Inform. Theory 52(5), 1990–2005 (2006)

    Article  MathSciNet  Google Scholar 

  7. Bölcskel, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46(12), 3256–3268 (1998)

    Article  Google Scholar 

  8. Candés, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities. Comm. Pure Appl. Math. 57(2), 219–266 (2004)

    Article  MathSciNet  Google Scholar 

  9. Casazza, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3, 543–557 (1997)

    Article  MathSciNet  Google Scholar 

  10. Casazza, P.G., Dilworth, S.J., Odell, E., Schlumprecht, Th, Zsák, A.: Coefficient quantization for frames in Banach spaces. J. Math. Anal. Appl. 348(1), 66–86 (2008)

    Article  MathSciNet  Google Scholar 

  11. Casazza, P.G., Freeman, D., Lynch, R.G.: Weaving Schauder frames. J. Approx. Theory 211, 42–60 (2016)

    Article  MathSciNet  Google Scholar 

  12. Casazza, P.G., Han, D., Larson, D.R.: Frames for Banach spaces. Contemp. Math. 247, 149–182 (1999)

    Article  MathSciNet  Google Scholar 

  13. Casazza, P.G., Kutyniok, G., Lammers, M.C.: Duality principles in frame theory. J. Fourier Anal. Appl. 10, 383–408 (2004)

    Article  MathSciNet  Google Scholar 

  14. Christensen, O.: an introduction to frames and Riesz bases. Birkhäuser, Basel (2016)

    MATH  Google Scholar 

  15. 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(2), 198–214 (2014)

    Article  MathSciNet  Google Scholar 

  16. Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nachr. 185(1), 33–47 (1997)

    Article  MathSciNet  Google Scholar 

  17. Christensen, O., Kim, H.O., Kim, R.Y.: On the duality principle by Casazza, Kutyniok, and Lammers. J. Fourier Anal. Appl. 17, 640–655 (2011)

    Article  MathSciNet  Google Scholar 

  18. Christensen, O., Xiao, X.C., Zhu, Y.C.: Characterizing R-duality in Banach spaces. Acta Math. Sinica Engl Ser. 29(1), 75–84 (2013)

    Article  MathSciNet  Google Scholar 

  19. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)

    Article  MathSciNet  Google Scholar 

  20. Daubechies, I., Landau, H.J., Landau, Z.: Gabor time-frequency lattices and the Wexler–Raz identity. J. Fourier Anal. Appl. 1(4), 437–478 (1995)

    Article  MathSciNet  Google Scholar 

  21. Daubechies, I.: The wavelet transform, time frequency localization and signal analysis. IEEE Trans. Inform. Theor. 36, 961–1005 (1990)

    Article  MathSciNet  Google Scholar 

  22. Devore, R. A., Jawerth, B., Lucier, B. J.: Image compression through wavelet transform coding. IEEE Trans. Inform. Theor. 38(2), 719–746 (1992)

  23. Diestel, J.: Seq. Ser. Banach spaces. Springer, New York (1984)

    Book  Google Scholar 

  24. Enayati, F., Asgari, M.S.: Duality properties for generalized frames. Banach J. Math. Anal. 11(4), 880–898 (2017)

    Article  MathSciNet  Google Scholar 

  25. Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. Rocky Mountain J. Math. 19(1), 113–126 (1989)

    Article  MathSciNet  Google Scholar 

  26. Frazier, M., Jawerth, B.: Decompositions of besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)

    Article  MathSciNet  Google Scholar 

  27. Gröchenig, K.: Describing functions: atomic decompositions versus frames. Monatshefte für Math. 112, 1–42 (1991)

    Article  MathSciNet  Google Scholar 

  28. Han, D., Larson, D.: Frames, bases and group representation. Mem. Am. Math. Soc. 147, (2000)

  29. Heath, R.W., Paulraj, A.J.: Linear dispersion codes for MIMO systems based on frame theory. IEEE Trans. Signal Process. 50(10), 2429–2441 (2002)

    Article  Google Scholar 

  30. Heil, C.: A basis theory primer. Birkhäuser, New York (2011)

    Book  Google Scholar 

  31. Janssen, A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)

    Article  MathSciNet  Google Scholar 

  32. Li, F., Li, P., Han, D.: Continuous framings for Banach spaces. J. Funct. Anal. 271(4), 992–1021 (2016)

    Article  MathSciNet  Google Scholar 

  33. Mallat, S.: A wavelet tour of signal processing. Academic Press, New York (1998)

    MATH  Google Scholar 

  34. Meyer, Y.: Principe d’incertitude, bases hilbertiennes et algébres d’opérateurs. Séminaire Bourbaki 662, 209–223 (1987)

    MATH  Google Scholar 

  35. Osgooei, E., Najati, A., Faroughi, M.H.: g-Riesz dual sequences for g-Bessel sequences. Asian Eur. J. Math. 7(3), 1450041 (2014)

    Article  MathSciNet  Google Scholar 

  36. Ron, A., Shen, Z.: Weyl-Heisenberg frames and Riesz bases in \(L_2(\mathbb{R}^d)\). Duke Math. J. 89(2), 237–282 (1997)

    Article  MathSciNet  Google Scholar 

  37. Singer, I.: Bases in Banach spaces-II. Springer, Berlin-Heidelberg, New York (1981)

    Book  Google Scholar 

  38. Speckbacher, M., Balazs, P.: Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups. J. Phys. A, Math. Theor. 48, 395201 (16pp) (2015)

  39. Stoeva, D.T., Christensen, O.: On various R-duals and the duality principle. Integr. Equ. Oper. Theory 84, 577–590 (2016)

    Article  MathSciNet  Google Scholar 

  40. Stoeva, D.T.: \(X_d\)-Frames in Banach spaces and their duals. Int. J. Pure Appl. Math. 52(1), 1–14 (2009)

    MathSciNet  MATH  Google Scholar 

  41. Takhteh, F., KHosravi, A.: R-duality in g-frames. Rocky Mountain J. Math. 47(2), 649–665 (2017)

  42. Xiao, X. M., Zhu, Y. C.: Duality principles of frames in Banach spaces. Acta Math. Sci. Ser. A Chin. Ed., 29(1), 94–102 (2009)

  43. Zhang, H., Zhang, J.: Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products. Appl. Comput. Harmon. Anal. 31(1), 1–25 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors’ work was partially supported by the Science and Research Branch of Islamic Azad University.

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Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Somayeh Hashemi Sanati & Mahdi Azhini

  2. Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, P. O. Box 13185/768, Tehran, Iran

    Mohammad Sadegh Asgari

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  1. Somayeh Hashemi Sanati
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  2. Mohammad Sadegh Asgari
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  3. Mahdi Azhini
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Correspondence to Mohammad Sadegh Asgari.

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Generalized framing and Riesz-dual sequences S. Hashemi Sanati, M. S. Asgari and M. Azhini.

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Hashemi Sanati, S., Asgari, M.S. & Azhini, M. Generalized framing and Riesz-dual sequences in Banach spaces. Bol. Soc. Mat. Mex. 27, 27 (2021). https://doi.org/10.1007/s40590-021-00336-0

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