Abstract
In this paper we mainly discuss the uniform excess of frames in finite dimensional Hilbert spaces. We discuss the relationship between uniform excess frames and full spark frames, give several methods to construct uniform excess frames. We also show that every full spark frame has the full range property and the intersection dependent property, and is linearly connected. At the same time we give a sufficient condition for two uniform excess frames can be woven.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexeev, B., Cahill, J., Mixon, D.G.: Full spark frames. J. Fourier Anal. Appl. 18, 1167–1194 (2012)
Bemrose, T., Casazza, P.G., Grochenig, K., Lammers, M.: Weaving frames. Oper. Matrices 10(4), 1093–1116 (2016)
Casazza, P.G., Kovacevic, J.: Equal-norm tight frames with erasures. Adv. Comput. Math. 392, 387–430 (2006)
Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)
Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25, 114–132 (2008)
Christensen, O., Powell, A.M., Xiao, X.C.: A note on finite dual frame pairs. Proc. Am. Math. Soc. 140(11), 3921–3930 (2012)
Copenhaver, M.S., Kim, Y.H., Logan, C., et al.: Maximum robustness and surgery of frames in finite dimensions. Linear Algebra Appl. 439, 1330–1339 (2013)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic fourier series. Trans. Math. Soc. 72, 341–366 (1952)
Christensen, O.: An Introduction to Frames and Riesz Bbases, 2nd edn. Birkhäuser, Boston (2015)
Găvruţa, L.: Frames for operators. Appl. Comput. Harm. Anal. 32, 139–144 (2012)
Găvruţa, P.: On the duality of fusion frames. J. Math. Anal. Appl. 333, 871–879 (2007)
Han, D.G., Kornelson, K., Larson, D., Weber, E.: Frames for Undergraduates. Student Mathematical Library, vol. 40. American Mathematical Society, Providence (2007)
Han, D.G., Sun, W.C.: Reconstruction of signals from frame coefficients with erasures at unknown locations. IEEE Trans. Inform. Theory 60, 4013–4025 (2014)
Khachiyan, L.: On the complexity of approximationg extremal determinants in matrices. J. Complex. 11, 138–153 (1995)
Narayan, S.K., Radzwion, E.L., Rimer, S.P., et al.: Robustness and surgery of frames. Linear Algebra Appl. 434, 1893–1901 (2011)
Pehlivan, S., Han, D.G., Mohapatra, R.: Linearly connected sequences and spectrally optimal dual frames for erasures. J. Func. Anal. 265, 2855–2876 (2013)
Sun, W.C.: G-frames and g-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Xiao, X.C., Zhu, Y.C., Găvruţa, L.: Some properties of K-frames in Hilbert spaces. Results Math. 63, 1243–1255 (2013)
Zhu, Y.C.: Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Math. Sin. (Engl. Ser.) 24(10), 1727–1736 (2008)
Acknowledgements
We thank the anonymous referees for valuable suggestions and comments on constructing uniform frames and testing for uniform excess being NP-hard, which lead to a significant improvement of our manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is partly supported by the Natural Science Foundation of Fujian Province, China (Grant No. 2016J01014), the Key funded research projects of higher institutions of Henan province (Grant No. 17A110015), and the projects of Xiamen University of Technology (Grant No. G2017005).
Rights and permissions
About this article
Cite this article
Xiao, X., Zhou, G. & Zhu, Y. Uniform Excess Frames in Hilbert Spaces. Results Math 73, 108 (2018). https://doi.org/10.1007/s00025-018-0871-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0871-0