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Balian–Low Type Theorems on Homogeneous Groups

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Abstract

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set {π(λ)g : λ ∈ Λ} for a discrete subset Λ ⊆ N/Z(N) forms a frame for Hπ, then its density satisfies the strict inequality D − (Λ) > dπ, where D − (Λ) is the lower Beurling density. An analogous density condition D+(Λ) < dπ holds for a Riesz sequence in Hπ contained in the orbit of (π,Hπ). The proof is based on a deformation theorem for coherent systems, a universality result for p-frames and p-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

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References

  1. A. Aldroubi, A. Baskakov, and I. Krishtal, Slanted matrices, Banach frames, and sampling, J. Funct. Anal., 255 (2008), 1667–1691.

    MathSciNet  MATH  Google Scholar 

  2. G. Ascensi, H. G. Feichtinger, and N. Kaiblinger, Dilation of the Weyl symbol and Balian-Low theorem, Trans. Amer. Math. Soc., 366 (2014), 3865–3880.

    MathSciNet  MATH  Google Scholar 

  3. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames. I. Theory, J. Fourier Anal. Appl., 12 (2006), 105–143.

    MathSciNet  MATH  Google Scholar 

  4. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames. II. Gabor systems, J. Fourier Anal. Appl., 12 (2006), 309–344.

    MathSciNet  MATH  Google Scholar 

  5. G. Battle, Heisenberg proof of the Balian-Low theorem, Lett. Math. Phys., 15 (1988), 175–177.

    MathSciNet  Google Scholar 

  6. A. Beurling, The Collected Works of Arne Beurling, Vol. 1, Complex Analysis, Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer, Contemporary Mathematicians, Birkhäuser Boston, Inc. (Boston, MA, 1989).

    MATH  Google Scholar 

  7. O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math., 26 (1996), 1289–1312.

    MathSciNet  MATH  Google Scholar 

  8. L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie groups and their Applications, Part I, Basic Theory and Examples, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press (Cambridge, 1990).

  9. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36 (1990), 961–1005.

    MathSciNet  MATH  Google Scholar 

  10. M. A. de Gosson, K. Gröchenig, and J. L. Romero, Stability of Gabor frames under small time Hamiltonian evolutions, Lett. Math. Phys., 106 (2016), 799–809.

    MathSciNet  MATH  Google Scholar 

  11. J. Dixmier and P. Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2), 102 (1978), 307–330.

    MathSciNet  MATH  Google Scholar 

  12. J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc., 76 (1970), 52–56.

    MathSciNet  MATH  Google Scholar 

  13. H. G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, (Budapest, 1980), Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland (Amsterdam, 1983), pp. 509–524.

    Google Scholar 

  14. H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal., 86 (1989), 307–340.

    MathSciNet  MATH  Google Scholar 

  15. H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math., 108 (1989), 129–148.

    MathSciNet  MATH  Google Scholar 

  16. H. G. Feichtinger and N. Kaiblinger, Varying the time-frequency lattice of Gabor frames, Trans. Amer. Math. Soc., 356 (2004), 2001–2023.

    MathSciNet  MATH  Google Scholar 

  17. V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, vol. 314, Birkhäuser/Springer (Cham, 2016).

  18. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, vol. 28, Princeton University Press (Princeton, N.J.); University of Tokyo Press (Tokyo, 1982).

  19. J. J. F. Fournier and J. Stewart, Amalgams of Lp and lq, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 1–21.

    Google Scholar 

  20. H. Führ and K. Gröchenig, Sampling theorems on locally compact groups from oscillation estimates, Math. Z., 255 (2007), 177–194.

    MathSciNet  MATH  Google Scholar 

  21. H. Führ, K. Gröchenig, A. Haimi, A. Klotz, and J. L. Romero, Density of sampling and interpolation in reproducing kernel Hilbert spaces, J. Lond. Math. Soc. (2), 96 (2017), 663–686.

    MathSciNet  MATH  Google Scholar 

  22. K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math., 112 (1991), 1–42.

    MathSciNet  MATH  Google Scholar 

  23. K. Gröchenig, The homogeneous approximation property and the comparison theorem for coherent frames, Sampl. Theory Signal Image Process., 7 (2008), 271–279.

    MathSciNet  MATH  Google Scholar 

  24. K. Gröchenig, Wiener's lemma: theme and variations. An introduction to spectral invariance, in: Four Short Courses on Harmonic Analysis, B. Forster and P. Massopust, eds., Appl. Num. Harm. Anal., Birkhäuser (Boston, 2010).

    Google Scholar 

  25. K. Gröchenig, J. Ortega-Cerdà, and J. L. Romero, Deformation of Gabor systems, Adv. Math., 277 (2015), 388–425.

    MathSciNet  MATH  Google Scholar 

  26. K. Gröchenig and M. Piotrowski, Molecules in coorbit spaces and boundedness of operators, Studia Math., 192 (2009), 61–77.

    MathSciNet  MATH  Google Scholar 

  27. K. Gröchenig and D. Rottensteiner, Orthonormal bases in the orbit of squareintegrable representations of nilpotent Lie groups, J. Funct. Anal., 275 (2018), 3338–3379.

    MathSciNet  MATH  Google Scholar 

  28. K. H. Gröchenig, H. Haimi, J. Ortega-Cerda, and J. L. Romero, Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions, J. Funct. Anal., 277 (2019), id 108282.

    MathSciNet  MATH  Google Scholar 

  29. A. Höfler, Necessary density conditions for frames on homogeneous groups, PhD thesis, Universität Wien (2014).

    Google Scholar 

  30. F. Holland, Harmonic analysis on amalgams of Lp and 1q, J. London Math. Soc. (2), 10 (1975), 295–305.

    MathSciNet  MATH  Google Scholar 

  31. R. W. Johnson, Homogeneous Lie algebras and expanding automorphisms, Proc. Amer. Math. Soc., 48 (1975), 292–296.

    MathSciNet  MATH  Google Scholar 

  32. H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117 (1967), 37–52.

    MathSciNet  MATH  Google Scholar 

  33. M. Mitkovsi and A. Ramirez, Density results for continuous frames, J. Fourier Anal. Appl., 26 (2020), Paper 56, 26 pp.

    MathSciNet  Google Scholar 

  34. C. C. Moore and J. A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc., 185 (1974), 445–462.

    MATH  Google Scholar 

  35. J. Ortega-Cerdà and K. Seip, Beurling-type density theorems for weighted Lp spaces of entire functions, J. Anal. Math., 75 (1988), 247–266.

    MATH  Google Scholar 

  36. V. Oussa, Frames arising from irreducible solvable actions. I, J. Funct. Anal., 274 (2018), 1202–1254.

    MathSciNet  MATH  Google Scholar 

  37. V. Oussa, Compactly supported bounded frames on Lie groups, J. Funct. Anal., 277 (2019), 1718–1762.

    MathSciNet  MATH  Google Scholar 

  38. J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2 (1995), 148–153.

    MathSciNet  MATH  Google Scholar 

  39. J. L. Romero, Surgery of spline-type and molecular frames, J. Fourier Anal. Appl., 17 (2011), 135–174.

    MathSciNet  MATH  Google Scholar 

  40. J. L. Romero, Characterization of coorbit spaces with phase-space covers, J. Funct. Anal., 262 (2012), 59–93.

    MathSciNet  MATH  Google Scholar 

  41. W. Rudin, Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc. (New York, 1991).

    MATH  Google Scholar 

  42. C. E. Shin and Q. Sun, Stability of localized operators, J. Funct. Anal., 256 (2009), 2417–2439.

    MathSciNet  MATH  Google Scholar 

  43. C. E. Shin and Q. Sun, Polynomial control on stability, inversion and powers of matrices on simple graphs, J. Funct. Anal., 276 (2019), 148–182.

    MathSciNet  MATH  Google Scholar 

  44. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in: Séminaire sur les Équations aux Dérivées Partielles (1994-1995), pages Exp. No. IV, 21, École Polytech. (Palaiseau, 1995).

    Google Scholar 

  45. Q. Sun, Wiener's lemma for infinite matrices, Trans. Amer. Math. Soc., 359 (2007), 3099–3123.

    MathSciNet  MATH  Google Scholar 

  46. R. Tessera, Left inverses of matrices with polynomial decay, J. Funct. Anal., 259 (2010), 2793–2813.

    MathSciNet  MATH  Google Scholar 

  47. J. A. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, vol. 142, American Mathematical Society (Providence, RI, 2007).

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria

    K. Gröchenig, J. L. Romero & J. T. Van Velthoven

  2. Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, A-1040, Vienna, Austria

    J. L. Romero

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    D. Rottensteiner

Authors
  1. K. Gröchenig
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  2. J. L. Romero
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  3. D. Rottensteiner
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  4. J. T. Van Velthoven
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Corresponding author

Correspondence to J. T. Van Velthoven.

Additional information

K. G. was partially supported by the project P31887-N32 of the Austrian Science Fund (FWF).

J. L.R. gratefully acknowledges support from the Austrian Science Fund (FWF): P 29462 and Y 1199 and from the WWTF grant INSIGHT (MA16-053).

D.R. was supported by the Austrian Science Fund (FWF) project I 3403.

J. v.V. acknowledges support from the Austrian Science Fund (FWF): P 29462-N35.

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Gröchenig, K., Romero, J.L., Rottensteiner, D. et al. Balian–Low Type Theorems on Homogeneous Groups. Anal Math 46, 483–515 (2020). https://doi.org/10.1007/s10476-020-0051-9

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