Uniform excess of g-frames

Ahmad Ahmadi; Abbas Askarizadeh

doi:10.1515/gmj-2023-2049

Article

Uniform excess of g-frames

Ahmad Ahmadi and Abbas Askarizadeh

Published/Copyright: July 25, 2023

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From the journal Georgian Mathematical Journal Volume 30 Issue 6

Abstract

Let { Λ i } i ∈ ℐ be a g-frame for 𝒰 with respect to { 𝒱 i } i ∈ ℐ , where 𝒰 and { 𝒱 i } i ∈ ℐ are Hilbert spaces. The excess of { Λ i } i ∈ ℐ is the largest cardinal number of the subsets 𝒥 of ℐ such that { Λ i } i ∈ ℐ ∖ 𝒥 is a g-frame for 𝒰 with respect to { 𝒱 i } i ∈ ℐ ∖ 𝒥 . In this paper, we consider the excess of a g-frame and introduce the concept of m-uniform excess for g-frames. Also, we present constructions of m-uniform excess g-frames. Finally, we investigate the relationship between m-uniform excess g-frames and woven g-frames.

Keywords: Uniform excess; full spark; woven frames

MSC 2020: 42C15; 46B28

Acknowledgements

The authors are grateful to the referees for their careful reading and useful comments.

References

[1] T. Bemrose, P. G. Casazza, K. Gröchenig, M. C. Lammers and R. G. Lynch, Weaving frames, Oper. Matrices 10 (2016), no. 4, 1093–1116. 10.7153/oam-10-61Search in Google Scholar

[2] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), no. 4, 389–427. 10.1006/acha.1997.0237Search in Google Scholar

[3] B. G. Bodmann, D. W. Kribs and V. I. Paulsen, Decoherence-insensitive quantum communication by optimal C * -encoding, IEEE Trans. Inform. Theory 53 (2007), no. 12, 4738–4749. 10.1109/TIT.2007.909105Search in Google Scholar

[4] P. G. Casazza, D. Han and D. R. Larson, Frames for Banach spaces, The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio 1999), Contemp. Math. 247, American Mathematical Society, Providence (1999), 149–182. 10.1090/conm/247/03801Search in Google Scholar

[5] P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (2008), no. 1, 114–132. 10.1016/j.acha.2007.10.001Search in Google Scholar

[6] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2003. 10.1007/978-0-8176-8224-8Search in Google Scholar

[7] C. E. D’Attellis and E. M. Fernández-Berdaguer, Wavelet Theory and Harmonic Analysis in Applied Sciences, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 1997. 10.1007/978-1-4612-2010-7Search in Google Scholar

[8] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271–1283. 10.1063/1.527388Search in Google Scholar

[9] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. 10.1090/S0002-9947-1952-0047179-6Search in Google Scholar

[10] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0003-1Search in Google Scholar

[11] D. Han and W. Sun, Reconstruction of signals from frame coefficients with erasures at unknown locations, IEEE Trans. Inform. Theory 60 (2014), no. 7, 4013–4025. 10.1109/TIT.2014.2320937Search in Google Scholar

[12] A. A. Hemmat and J.-P. Gabardo, Properties of oblique dual frames in shift-invariant systems, J. Math. Anal. Appl. 356 (2009), no. 1, 346–354. 10.1016/j.jmaa.2009.01.044Search in Google Scholar

[13] A. J. E. M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl. 83 (1981), no. 2, 377–394. 10.1016/0022-247X(81)90130-XSearch in Google Scholar

[14] A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), no. 5, 720–731. 10.1063/1.525426Search in Google Scholar

[15] A. Khosravi and K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl. 342 (2008), no. 2, 1068–1083. 10.1016/j.jmaa.2008.01.002Search in Google Scholar

[16] D. Li, J. Leng, T. Huang and X. Li, On weaving g-frames for Hilbert spaces, Complex Anal. Oper. Theory 14 (2020), no. 2, Paper No. 33, 10.1007/s11785-020-00991-7. 10.1007/s11785-020-00991-7Search in Google Scholar

[17] A. Najati, M. H. Faroughi and A. Rahimi, g-frames and stability of g-frames in Hilbert spaces, Methods Funct. Anal. Topology 14 (2008), no. 3, 271–286. Search in Google Scholar

[18] A. Najati and A. Rahimi, Generalized frames in Hilbert spaces, Bull. Iranian Math. Soc. 35 (2009), no. 1, 97–109. Search in Google Scholar

[19] A. Ron and Z. Shen, Weyl–Heisenberg frames and Riesz bases in L 2 ⁢ ( 𝐑 d ) , Duke Math. J. 89 (1997), no. 2, 237–282. 10.1215/S0012-7094-97-08913-4Search in Google Scholar

[20] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), no. 1, 437–452. 10.1016/j.jmaa.2005.09.039Search in Google Scholar

[21] W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326 (2007), no. 2, 858–868. 10.1016/j.jmaa.2006.03.043Search in Google Scholar

[22] X. Xiao, G. Zhou and Y. Zhu, Uniform excess frames in Hilbert spaces, Results Math. 73 (2018), no. 3, Paper No. 108, /10.1007/s00025-018-0871-0. /10.1007/s00025-018-0871-0Search in Google Scholar

[23] R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure Appl. Math. 93, Academic Press, New York, 1980. Search in Google Scholar

[24] L. Zang, W. Sun and D. Chen, Excess of a class of g-frames, J. Math. Anal. Appl. 352 (2009), no. 2, 711–717. 10.1016/j.jmaa.2008.11.030Search in Google Scholar

Received: 2022-12-08

Revised: 2023-01-19

Accepted: 2023-01-25

Published Online: 2023-07-25

Published in Print: 2023-12-01

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Articles in the same Issue

https://doi.org/10.1515/gmj-2023-2049

Keywords for this article

Uniform excess; full spark; woven frames