Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, U. Molter, Invariance of a shift-invariant space. J. Fourier Anal. Appl. 16(1), 60–75 (2010)
A. Aldroubi, Q. Sun, H. Wang, Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces. Appl. Comput. Harmon. Anal. 30 (2011), no. 3, 337–347.
M. Anastasio, C. Cabrelli, V. Paternostro, Invariance of a shift-invariant space in several variables. Complex Anal. Oper. Theory 5(4), 1031–1050 (2011)
R. Balian, Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R. Acad. Sci. 292(20), 1357–1362 (1981)
J. Benedetto, C. Heil, D. Walnut, Differentiation and the Balian-Low theorem. J. Fourier Anal. Appl. 1(4), 355–402 (1995)
J. Benedetto, W. Czaja, P. Gadzinski, A.M. Powell, The Balian-Low theorem and regularity of Gabor systems. J. Geom. Anal. 13(2), 239–254 (2003)
C. Cabrelli, U. Molter, G. Pfander, Time-frequency shift invariance and the amalgam Balian-Low theorem. Appl. Comput. Harmon. Anal. 41(3), 677–691 (2016)
A. Caragea, D.G. Lee, F. Philipp, F. Voigtlaender, A quantitative subspace Balian-Low theorem. Appl. Comput. Harmon. Anal. 55, 368–404 (2021)
I. Daubechies, A.J.E.M. Janssen, Two theorems on lattice expansions. IEEE Trans. Inf. Theory 39(1), 3–6 (1993)
J.-P. Gabardo, D. Han, Balian-Low phenomenon for subspace Gabor frames. J. Math. Phys. 45(8), 3362–3378 (2004)
S.Z. Gautam, A critical-exponent Balian-Low theorem. Math. Res. Lett. 15(3), 471–483 (2008)
L. Grafakos, Classical Fourier Analysis (Springer, New York, 2014)
K. Gröchenig, J.L. Romero, D. Rottensteiner, J.T. van Velthoven, Balian-Low type theorems on homogeneous groups. Anal. Math. 46(3), 483–515 (2020)
D. Hardin, M. Northington, A.M. Powell, A sharp Balian-Low uncertainty principle for shift-invariant spaces. Appl. Comput. Harmon. Anal. 44(2), 294–311 (2018)
C. Heil, A Basis Theory Primer. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2011)
C. Heil, A.M. Powell, Regularity for complete and minimal Gabor systems on a lattice. Ill. J. Math. 53(4), 1077–1094 (2009)
E. Hernandez, H. Sikic, G. Weiss, E. Wilson, On the properties of the integer translates of a square integrable function, Harmonic Analysis and Partial Differential Equations. Contemporary Mathematics, vol. 505 (American Mathematical Society, Providence, RI, 2010), pp. 233–249
R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973)
L. Hörmander, Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)
F. Low, Complete sets of wave packets, in A Passion for Physics—Essays in Honor of Geoffrey Chew, ed. by C. DeTar et al. (World Scientific, Singapore, 1985), pp. 17–22
S. Nitzan, J.-F. Olsen, From exact systems to Riesz bases in the Balian-Low theorem. J. Fourier Anal. Appl. 17(4), 567–603 (2011)
M. Northington, Uncertainty principles for Fourier multipliers. J. Fourier Anal. Appl. 26(5), Paper No. 76, 38 pp. (2020)
E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, 1970)
R. Tessera, H. Wang, Uncertainty principles in finitely generated shift-invariant spaces with additional invariance. J. Math. Anal. Appl. 410(1), 134–143 (2014)
A. Zygmund, Trigomometric Series, 3rd edn. (Cambridge University Press, New York, 2002)
Acknowledgements
The results in this chapter should be considered joint work with Shahaf Nitzan and Michael Northington in connection with the collaborative work published in [22] and during visits at Vanderbilt University, Kent State University, and the Georgia Institute of Technology. The author thanks Chris Heil for valuable discussions on the Balian-Low theorem and Dechao Zheng for sharing his expertise on \(\mathcal {A}_p\) weights.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Powell, A.M. (2023). The Balian-Low Theorem for \((C_q)\)-Systems in Shift-Invariant Spaces. In: Casey, S.D., Dodson, M.M., Ferreira, P.J.S.G., Zayed, A. (eds) Sampling, Approximation, and Signal Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-41130-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-031-41130-4_6
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-41129-8
Online ISBN: 978-3-031-41130-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)