Coherent systems over approximate lattices in amenable groups

Enstad, Ulrik; van Velthoven, Jordy Timo

doi:10.5802/aif.3687

Coherent systems over approximate lattices in amenable groups
[Systèmes cohérents associés aux réseaux approximatifs dans des groupes moyennables]

Enstad, Ulrik ¹ ; van Velthoven, Jordy Timo ²

¹ Department of Mathematics University of Oslo Moltke Moes vei 35 0851 Oslo (Norway)
² Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna (Austria)

Annales de l'Institut Fourier, Online first, 23 p.

Résumé
Abstract

Soit $G$ un groupe moyennable à base dénombrable et $Λ$ un réseau uniforme $k$ -approximatif dans $G$ . Pour une représentation projective de la série discète $(π, ℋ_{π})$ de $G$ de dimension formelle $d_{π} > 0$ , nous prouvons que la condition $D^{-} (Λ) \geq d_{π} / k$ est nécessaire pour que le système cohérent $π (Λ) g$ soit complet dans $ℋ_{π}$ . En outre, nous montrons que si le système $π (Λ^{2}) g$ est minimal, alors $D^{+} (Λ^{2}) \leq d_{π} k$ . Ces deux conditions nécessaires sont nouvelles, même pour des systèmes de Gabor dans $L^{2} (ℝ)$ et permettent de reprouver des théorèmes de densité stricte. Comme application de cette approche, nous obtenons également des conditions de densité pour les repères cohérents et les suites de Riesz associés à des ensembles discrets arbitraires. Tous nos résultats sont valables pour des groupes unimodulaires à croissance exponentielle.

Let $G$ be a second-countable amenable group with a uniform $k$ -approximate lattice $Λ$ . For a projective discrete series representation $(π, ℋ_{π})$ of $G$ of formal degree $d_{π} > 0$ , we show that $D^{-} (Λ) \geq d_{π} / k$ is necessary for the coherent system $π (Λ) g$ to be complete in $ℋ_{π}$ . In addition, we show that if $π (Λ^{2}) g$ is minimal, then $D^{+} (Λ^{2}) \leq d_{π} k$ . Both necessary conditions recover sharp density theorems for uniform lattices and are new even for Gabor systems in $L^{2} (ℝ)$ . As an application of the approach, we also obtain necessary density conditions for coherent frames and Riesz sequences associated to general discrete sets. All results are valid for amenable unimodular groups of possibly exponential growth.

Reçu le : 2022-08-15
Révisé le : 2022-10-31
Accepté le : 2023-10-02
Première publication : 2025-02-10

DOI : 10.5802/aif.3687

Classification : 22D25, 22E27, 42C30, 42C40
Keywords: Approximate lattice, complete systems, density condition, discrete series, frame
Mots-clés : Condition de densité, repères, réseau uniforme approximatif, série discète, système complet

Affiliations des auteurs :

Enstad, Ulrik ¹ ; van Velthoven, Jordy Timo ²

¹ Department of Mathematics University of Oslo Moltke Moes vei 35 0851 Oslo (Norway)
² Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna (Austria)

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     title = {Coherent systems over approximate lattices in amenable groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3687},
     language = {en},
     note = {Online first},
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Enstad, Ulrik; van Velthoven, Jordy Timo. Coherent systems over approximate lattices in amenable groups. Annales de l'Institut Fourier, Online first, 23 p.

Bibliographie
Cité par

[1] Ahn, Andrew; Clark, William; Nitzan, Shahaf; Sullivan, Joseph Density of Gabor systems via the short time Fourier transform, J. Fourier Anal. Appl., Volume 24 (2018) no. 3, pp. 699-718 | DOI | MR | Zbl

[2] Ascensi, Gerard; Lyubarskii, Yurii; Seip, Kristian Phase space distribution of Gabor expansions, Appl. Comput. Harmon. Anal., Volume 26 (2009) no. 2, pp. 277-282 | DOI | MR | Zbl

[3] Baake, Michael; Grimm, Uwe Aperiodic order. Volume 1. A mathematical invitation, Encyclopedia of Mathematics and Its Applications, 149, Cambridge University Press, 2013, xvi+531 pages | DOI | MR | Zbl

[4] Bekka, Bachir Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., Volume 10 (2004) no. 4, pp. 325-349 | DOI | MR | Zbl

[5] Belov, Yurii; Borichev, Alexander; Kuznetsov, Alexander Upper and lower densities of Gabor Gaussian systems, Appl. Comput. Harmon. Anal., Volume 49 (2020) no. 2, pp. 438-450 | DOI | MR | Zbl

[6] Björklund, Michael; Hartnick, Tobias Approximate lattices, Duke Math. J., Volume 167 (2018) no. 15, pp. 2903-2964 | DOI | MR | Zbl

[7] Björklund, Michael; Hartnick, Tobias; Pogorzelski, Felix Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets, Proc. Lond. Math. Soc., Volume 116 (2018) no. 4, pp. 957-996 | DOI | MR | Zbl

[8] Breuillard, Emmanuel Geometry of locally compact groups of polynomial growth and shape of large balls, Groups Geom. Dyn., Volume 8 (2014) no. 3, pp. 669-732 | DOI | MR | Zbl

[9] Breuillard, Emmanuel; Green, Ben; Tao, Terence The structure of approximate groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 116 (2012), pp. 115-221 | DOI | Numdam | MR | Zbl

[10] Dixmier, Jacques $C^{*}$ -algebras, North-Holland Mathematical Library, 15, Elsevier, 1977 | MR | Zbl

[11] Enstad, Ulrik The density theorem for projective representations via twisted group von Neumann algebras, J. Math. Anal. Appl., Volume 511 (2022) no. 2, 126072, 25 pages | DOI | MR | Zbl

[12] Enstad, Ulrik; Raum, Sven A dynamical approach to non-uniform density theorems for coherent systems (2022) (to appear in Trans. Am. Math. Soc., https://arxiv.org/abs/2207.05125) | DOI

[13] Enstad, Ulrik; van Velthoven, Jordy Timo On sufficient density conditions for lattice orbits of relative discrete series, Arch. Math., Volume 119 (2022) no. 3, pp. 279-291 | DOI | MR | Zbl

[14] Führ, Hartmut; Gröchenig, Karlheinz; Haimi, Antti; Klotz, Andreas; Romero, José Luis Density of sampling and interpolation in reproducing kernel Hilbert spaces, J. Lond. Math. Soc., Volume 96 (2017) no. 3, pp. 663-686 | DOI | MR | Zbl

[15] Gaal, Steven A. Linear analysis and representation theory, Grundlehren der Mathematischen Wissenschaften, 198, Springer, 1973, ix+688 pages | DOI | MR | Zbl

[16] Gabardo, Jean Pierre; Han, Deguang Frame representations for group-like unitary operator systems, J. Oper. Theory, Volume 49 (2003) no. 2, pp. 223-244 | MR | Zbl

[17] Gröchenig, Karlheinz Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and algorithms. Theory and applications (Applied and Numerical Harmonic Analysis), Birkhäuser, 1998, p. 211-231, 453–488 | MR | Zbl

[18] Gröchenig, Karlheinz The homogeneous approximation property and the comparison theorem for coherent frames, Sampl. Theory Signal Image Process., Volume 7 (2008) no. 3, pp. 271-279 stsip.org/vol07/no3/vol07no3pp271-279.html | DOI | MR | Zbl

[19] Heil, Christopher History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl., Volume 13 (2007) no. 2, pp. 113-166 | DOI | MR | Zbl

[20] Jenkins, J. W. Growth of connected locally compact groups, J. Funct. Anal., Volume 12 (1973), pp. 113-127 | DOI | MR | Zbl

[21] Machado, Simon Approximate lattices and Meyer sets in nilpotent Lie groups, Discrete Anal., Volume 2020 (2020), 1, 18 pages | DOI | MR | Zbl

[22] Machado, Simon Infinite approximate subgroups of soluble Lie groups, Math. Ann., Volume 382 (2022) no. 1-2, pp. 285-301 | DOI | MR | Zbl

[23] Machado, Simon Closed approximate subgroups: Compactness, amenability and approximate lattices (2023) (https://arxiv.org/abs/2011.01829)

[24] Meyer, Yves Algebraic numbers and harmonic analysis, North-Holland Mathematical Library, 2, Elsevier, 1972, x+274 pages | MR | Zbl

[25] Mitkovski, Mishko; Ramirez, Aaron Density results for continuous frames, J. Fourier Anal. Appl., Volume 26 (2020) no. 4, 56, 26 pages | DOI | MR | Zbl

[26] Nitzan, Shahaf; Olevskii, Alexander Revisiting Landau’s density theorems for Paley-Wiener spaces, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 9-10, pp. 509-512 | DOI | MR | Zbl

[27] Ornstein, Donald S.; Weiss, Benjamin Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., Volume 48 (1987), pp. 1-141 | DOI | MR | Zbl

[28] Pogorzelski, Felix; Richard, Christoph; Strungaru, Nicolae Leptin densities in amenable groups, J. Fourier Anal. Appl., Volume 28 (2022) no. 6, 85, 36 pages | DOI | MR

[29] Pogorzelski, Felix; Schwarzenberger, Fabian A Banach space-valued ergodic theorem for amenable groups and applications, J. Anal. Math., Volume 130 (2016), pp. 19-69 | DOI | MR | Zbl

[30] Ramanathan, Jayakumar; Steger, Tim Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., Volume 2 (1995) no. 2, pp. 148-153 | DOI | MR | Zbl

[31] Romero, Eduardo A complete Gabor system of zero Beurling density, Sampl. Theory Signal Image Process., Volume 3 (2004) no. 3, pp. 299-304 | DOI | MR | Zbl

[32] Romero, José Luis; van Velthoven, Jordy Timo The density theorem for discrete series representations restricted to lattices, Expo. Math., Volume 40 (2022) no. 2, pp. 265-301 | DOI | MR | Zbl

[33] Rosenberg, Jonathan Square-integrable factor representations of locally compact groups, Trans. Am. Math. Soc., Volume 237 (1978), pp. 1-33 | DOI | MR | Zbl

[34] Schlottmann, Martin Generalized model sets and dynamical systems, Directions in mathematical quasicrystals (CRM Monograph Series), Volume 13, American Mathematical Society, 2000, pp. 143-159 | MR | Zbl

[35] Tao, Terence Product set estimates for non-commutative groups, Combinatorica, Volume 28 (2008) no. 5, pp. 547-594 | DOI | MR | Zbl

[36] Tessera, Romain Volume of spheres in doubling metric measured spaces and in groups of polynomial growth, Bull. Soc. Math. Fr., Volume 135 (2007) no. 1, pp. 47-64 smf4.emath.fr/en/publications/bulletin/135/html/smf_bull_135_47-64.html | DOI | Numdam | MR | Zbl

[37] Wang, Yang Sparse complete Gabor systems on a lattice, Appl. Comput. Harmon. Anal., Volume 16 (2004) no. 1, pp. 60-67 | DOI | MR | Zbl

[38] Zorin-Kranich, Pavel Return times theorem for amenable groups, Isr. J. Math., Volume 204 (2014), pp. 85-96 | DOI | MR | Zbl

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