# zbMATH — the first resource for mathematics

A sharp Balian-Low uncertainty principle for shift-invariant spaces. (English) Zbl 1391.42036
The authors prove a sharp version of the Balian-Low theorem for the generators of finitely generated shift-invariant spaces. For singly generated shift-invariant spaces their main results take the following form: 1.(Corollary 1.4) Fix lattices $$\Lambda,\Gamma\subset\mathbb{R}^d$$ with $$\Lambda\subset\Gamma$$ and with the index of the lattice $$\Lambda$$ in $$\Gamma$$ greater than 1 ($$[\Gamma\, : \,\Lambda]>1).$$ Suppose $$f\in L^2(\mathbb{R}^d),$$ $$\| f\|_2\neq 0,$$ and $$\mathcal{T}^{\Lambda}(f)=\{f(\cdot -\lambda)\, :\,\lambda\in\Lambda\}$$ forms a frame for the closed linear span $$V^{\Lambda}(f)$$ of $$\mathcal{T}^{\Lambda}(f)$$ in $$L^2(\mathbb{R}^d).$$ If $$V^{\Lambda}(f)$$ is $$\Gamma -$$ invariant, then $$\int_{L^2(\mathbb{R}^d)}|x||f(x)|^2dx=\infty.$$ 2.(Corollary 1.6) Fix a lattice $$\Lambda\subset\mathbb{R}^d.$$ Suppose $$f\in L^2(\mathbb{R}^d)$$ with $$\| f\|_2\neq 0.$$ If $$\mathcal{T}^{\Lambda}(f)$$ is a frame for $$V^{\Lambda}(f) ,$$ but is not a Riesz basis for $$V^{\Lambda}(f),$$ then $$\int_{L^2(\mathbb{R}^d)}|x||f(x)|^2dx=\infty.$$ The main results provide an absolutely sharp improvement of the best previously existing results in the literature in this direction.

##### MSC:
 42C15 General harmonic expansions, frames
Full Text:
##### References:
 [1] Aldroubi, A.; Cabrelli, C.; Heil, C.; Kornelson, K.; Molter, U., Invariance of a shift-invariant space, J. Fourier Anal. Appl., 16, 1, 60-75, (2010) · Zbl 1194.42042 [2] Aldroubi, A.; Sun, Q.; Wang, H., Uncertainty principles and Balian-low type theorems in principal shift-invariant spaces, Appl. Comput. Harmon. Anal., 30, 3, 337-347, (2011) · Zbl 1215.42041 [3] Anastasio, M.; Cabrelli, C.; Paternostro, V., Invariance of a shift-invariant space in several variables, Complex Anal. Oper. Theory, 5, 4, 1031-1050, (2011) · Zbl 1281.46027 [4] Ascensi, G.; Feichtinger, H.; Kaiblinger, N., Dilation of the Weyl symbol and Balian-low theorem, Trans. Amer. Math. Soc., 366, 7, 3865-3880, (2014) · Zbl 1298.47059 [5] Balan, R.; Daubechies, I., Optimal stochastic encoding and approximation schemes using Weyl-Heisenberg sets, (Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis, (2003), Birkhäuser Boston, MA), 259-320 · Zbl 1033.94001 [6] Balian, R., Un principe d’incertitude fort en théorie du signal ou en mécanique quantique, C. R. Acad. Sci., 292, 20, 1357-1362, (1981) [7] Bastys, A., Translation invariance of orthogonal multiresolution analyses of $$L^2(\mathbb{R})$$, Appl. Comput. Harmon. Anal., 9, 128-145, (2000) · Zbl 0979.42017 [8] Battle, G., Heisenberg proof of the Balian-low theorem, Lett. Math. Phys., 15, 175-177, (1988) [9] Benedetto, J. J.; Czaja, W.; Gadziński, P.; Powell, A. M., The Balian-low theorem and regularity of Gabor systems, J. Geom. Anal., 13, 2, 239-254, (2003) · Zbl 1037.42030 [10] Benedetto, J. J.; Czaja, W.; Maltsev, A., The Balian-low theorem for the symplectic form on $$\mathbb{R}^{2 d}$$, J. Math. Phys., 44, 4, 1735-1750, (2003) · Zbl 1062.42026 [11] Benedetto, J. J.; Czaja, W.; Powell, A. M., An optimal example for the Balian-low uncertainty principle, SIAM J. Math. Anal., 38, 1, 333-345, (2006) · Zbl 1142.42013 [12] Benedetto, J. J.; Czaja, W.; Powell, A. M.; Sterbenz, J., An endpoint $$(1, \infty)$$ Balian-low theorem, Math. Res. Lett., 13, 2-4, 467-474, (2006) · Zbl 1226.42022 [13] Benedetto, J. J.; Heil, C.; Walnut, D., Differentiation and the Balian-low theorem, J. Fourier Anal. Appl., 1, 4, 355-402, (1995) · Zbl 0887.42026 [14] Benedetto, J. J.; Li, S., The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 5, 4, 389-427, (1998) · Zbl 0915.42029 [15] Benyi, A.; Oh, T., The Sobolev inequality on the torus revisited, Publ. Math. Debrecen, 83, 3, 359-374, (2013) · Zbl 1313.42081 [16] Bhatia, R., Matrix analysis, (1997), Springer New York [17] Bourgain, J., Remark on the uncertainty principle for Hilbertian basis, J. Funct. Anal., 79, 1, 136-143, (1988) · Zbl 0656.46016 [18] Bourgain, J.; Brezis, H.; Mironescu, P., Lifting in Sobolev spaces, J. Anal. Math., 80, 37-86, (2000) · Zbl 0967.46026 [19] Bownik, M., The structure of shift-invariant subspaces of $$L^2(\mathbb{R}^n)$$, J. Funct. Anal., 177, 282-309, (2000) · Zbl 0986.46018 [20] Brezis, H.; Li, Y.; Mironescu, P.; Nirenberg, L., Degree and Sobolev spaces, Topol. Methods Nonlinear Anal., 13, 181-190, (1999) · Zbl 0956.46024 [21] Casazza, P.; Lammers, M., Bracket products for Weyl-Heisenberg frames, (Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis, (2003), Birkhäuser Boston), 71-98 · Zbl 1027.42027 [22] Christensen, O., An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, (2003), Birkhäuser Boston · Zbl 1017.42022 [23] Czaja, W.; Powell, A. M., Recent developments in the Balian-low theorem, (Harmonic Analysis and Applications, Applied and Numerical Harmonic Analysis, (2006), Birkäuser Boston), 79-100 · Zbl 1129.42410 [24] Daubechies, I.; Janssen, A. J.E. M., Two theorems on lattice expansions, IEEE Trans. Inform. Theory, 39, 1, 3-6, (1993) · Zbl 0764.42018 [25] de Boor, C.; DeVore, R. A.; Ron, A., The structure of finitely generated shift-invariant spaces in $$L_2(\mathbb{R}^d)$$, J. Funct. Anal., 119, 1, 37-78, (1994) · Zbl 0806.46030 [26] Folland, G. B.; Sitaram, A., The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3, 3, 207-238, (1997) · Zbl 0885.42006 [27] Gabardo, J.-P.; Han, D., Balian-low phenomenon for subspace Gabor frames, J. Math. Phys., 45, 8, 3362-3378, (2004) · Zbl 1071.42022 [28] Gautam, S. Z., A critical exponent Balian-low theorem, Math. Res. Lett., 15, 3, 471-483, (2008) · Zbl 1268.42056 [29] K. Gröchenig, personal communication. [30] Gröchenig, K.; Malinnikova, E., Phase space localization of Riesz bases for $$L^2(\mathbb{R}^d)$$, Rev. Mat. Iberoam., 29, 1, 115-134, (2013) · Zbl 1262.81081 [31] Gröchenig, K.; Han, D.; Heil, C.; Kutyniok, G., The Balian-low theorem for symplectic lattices in higher dimensions, Appl. Comput. Harmon. Anal., 13, 2, 169-176, (2002) · Zbl 1017.42027 [32] Havin, V.; Jöricke, B., The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 28, (1994), Spring-Verlag Berlin · Zbl 0827.42001 [33] Heil, C., History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl., 13, 2, 113-166, (2007) · Zbl 1133.42043 [34] Heil, C.; Powell, A. M., Gabor Schauder bases and the Balian-low theorem, J. Math. Phys., 47, 11, (2006) · Zbl 1112.42004 [35] Heil, C.; Powell, A. M., Regularity for complete and minimal Gabor systems on a lattice, Illinois J. Math., 53, 4, 1077-1094, (2009) · Zbl 1207.42025 [36] Hernández, E.; Šikić, H.; Weiss, G.; Wilson, E., On the properties of the integer translates of a square integrable function, Contemp. Math., 505, 233-249, (2010) · Zbl 1204.47009 [37] Hogan, J.; Lakey, J., Non-translation-invariance and the synchronization problem in wavelet sampling, Acta Appl. Math., 107, 373-398, (2009) · Zbl 1175.94061 [38] Hogan, J.; Lakey, J., Non-translation-invariance in principal shift-invariant spaces, (Advances in Analysis, (2005), World Scientific Publishing Hackensack, NJ), 471-485 · Zbl 1090.94013 [39] Janssen, A. J.E. M., A decay result for certain windows generating orthogonal Gabor bases, J. Fourier Anal. Appl., 14, 1-15, (2008) · Zbl 1268.42058 [40] Lammers, M., The finite Balian-low conjecture, (Proceedings of the 11th International Conference on Sampling Theory and Applications, Washington DC, May 25-29, 2015, (2015)), 25-29, 4 pages [41] Low, F., Complete sets of wave packets, (DeTar, C.; etal., A Passion for Physics—Essays in Honor of Geoffrey Chew, (1985), World Scientific Singapore), 17-22 [42] Nitzan, S.; Olsen, J.-F., From exact systems to Riesz bases in the Balian-low theorem, J. Fourier Anal. Appl., 17, 4, 567-603, (2011) · Zbl 1227.42035 [43] Nitzan, S.; Olsen, J.-F., A quantitative Balian-low theorem, J. Fourier Anal. Appl., 19, 5, 1078-1092, (2013) · Zbl 1304.42079 [44] Stein, E., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501 [45] Tessera, R.; Wang, H., Uncertainty principles in finitely generated shift-invariant spaces with additional invariance, J. Math. Anal. Appl., 410, 134-143, (2014) · Zbl 1320.42031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.