Time-frequency analysis on the adeles over the rationals

Ulrik B.R. Enstad; Mads S. Jakobsen; Franz Luef

doi:10.1016/j.crma.2018.12.004

Harmonic analysis

Time-frequency analysis on the adeles over the rationals
[Analyse des temps-fréquences sur l'anneau des adèles des rationnels]

Présenté par : the Editorial Board

Ulrik B.R. Enstad ¹ ; Mads S. Jakobsen ² ; Franz Luef ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 188-199.

Abstract (VO)
Résumé

We show that the construction of Gabor frames in $L^{2} (R)$ with generators in $S_{0} (R)$ and with respect to time-frequency shifts from a rectangular lattice $α Z \times β Z$ is equivalent to the construction of certain Gabor frames for $L^{2}$ over the adeles over the rationals and the group $R \times Q_{p}$ . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $R \times Q_{p}$ with the construction of certain Heisenberg modules.

Nous montrons que la construction de trames (ou repères) de Gabor de $L^{2} (R)$ avec générateurs dans $S_{0} (R)$ et des décalages de temps-fréquence dans un réseau rectangulaire $α Z \times β Z$ est équivalente à la construction de certaines trames de Gabor pour $L^{2}$ sur les adèles des rationnels avec le groupe $R \times Q_{p}$ . Nous analysons également les relations entre la construction de trames de Gabor sur les adèles et sur $R \times Q_{p}$ et la construction de certains modules de Heisenberg.

Reçu le : 2018-07-27
Accepté le : 2019-01-07
Publié le : 2019-01-28

DOI : 10.1016/j.crma.2018.12.004

Affiliations des auteurs :

Ulrik B.R. Enstad ¹ ; Mads S. Jakobsen ² ; Franz Luef ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

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     title = {Time-frequency analysis on the adeles over the rationals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {188--199},
     publisher = {Elsevier},
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     number = {2},
     year = {2019},
     doi = {10.1016/j.crma.2018.12.004},
     language = {en},
}

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Ulrik B.R. Enstad; Mads S. Jakobsen; Franz Luef. Time-frequency analysis on the adeles over the rationals. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 188-199. doi : 10.1016/j.crma.2018.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.12.004/

Bibliographie
Cité par

[1] O. Ahmad; F.A. Shah; N.A. Sheik Gabor frames on non-Archimedean fields, Int. J. Geom. Methods Mod. Phys., Volume 15 (2018) no. 05

[2] S. Albeverio; S. Evdokimov; M. Skopina p-adic multiresolution analysis and wavelet frames, Fourier Anal. Appl., Volume 16 (2010) no. 5, pp. 693-714

[3] J.J. Benedetto, R.L. Benedetto, Frames of translates for number-theoretic groups, ArXiv e-prints, 2018.

[4] O. Christensen An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Basel, 2016

[5] O. Christensen; S.S. Goh Fourier-like frames on locally compact Abelian groups, J. Approx. Theory, Volume 192 (2015), pp. 82-101

[6] X.-R. Dai; Q. Sun The abc-problem for Gabor systems, Mem. Amer. Math. Soc., Volume 244 (2016) no. 1152 (ix+99)

[7] A. Deitmar Automorphe Formen, Springer, Heidelberg, Germany, 2010

[8] H.G. Feichtinger On a new Segal algebra, Monatshefte Math., Volume 92 (1981), pp. 269-289

[9] H.G. Feichtinger Modulation Spaces on Locally Compact Abelian Groups, University of Vienna, January 1983 (Technical report)

[10] H.G. Feichtinger Modulation spaces of locally compact Abelian groups, Chennai, January 2002 (R. Radha; M. Krishna; S. Thangavelu, eds.), Allied Publishers, New Delhi (2003), pp. 1-56

[11] H.G. Feichtinger Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process., Int. J., Volume 5 (2006) no. 2, pp. 109-140

[12] H.G. Feichtinger; G. Zimmermann A Banach space of test functions for Gabor analysis (H.G. Feichtinger; T. Strohmer, eds.), Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 1998, pp. 123-170

[13] K. Gröchenig Aspects of Gabor analysis on locally compact Abelian groups (H.G. Feichtinger; T. Strohmer, eds.), Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, USA, 1998, pp. 123-170

[14] K. Gröchenig Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, USA, 2001

[15] K. Gröchenig A pedestrian's approach to pseudodifferential operators (C. Heil, ed.), Harmonic Analysis and Applications, Volume in Honor of John J. Benedetto's 65th Birthday, Birkhäuser Boston, Boston, MA, 2006, pp. 139-169

[16] K. Gröchenig; M. Leinert Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., Volume 17 (2004), pp. 1-18

[17] K. Gröchenig; J.L. Romero; J. Stöckler Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions, Invent. Math., Volume 211 (2018) no. 3, pp. 1119-1148

[18] K. Gröchenig; J. Stöckler Gabor frames and totally positive functions, Duke Math. J., Volume 162 (2013) no. 6, pp. 1003-1031

[19] K. Gröchenig; T. Strohmer Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class, J. Reine Angew. Math., Volume 613 (2007), pp. 121-146

[20] V.P. Havin; N.K. Nikol'skij Commutative Harmonic Analysis II. Group Methods in Commutative Harmonic Analysis, Springer, Berlin, 1998 (Translated from Russian)

[21] E. Hewitt; K.A. Ross Abstract Harmonic Analysis I, Grundlehren Math. Wiss., vol. 115, Springer, Berlin, 1963

[22] M.S. Jakobsen On a (no longer) new Segal algebra: a review of the Feichtinger algebra, J. Fourier Anal. Appl., Volume 24 (2018) no. 6, pp. 1579-1660

[23] M.S. Jakobsen; J. Lemvig Density and duality theorems for regular Gabor frames, J. Funct. Anal., Volume 270 (2016) no. 1, pp. 229-263

[24] M.S. Jakobsen, F. Luef, Duality of Gabor frames and Heisenberg modules, ArXiv e-prints, 2018.

[25] A.J.E.M. Janssen Zak transforms with few zeros and the tie (H.G. Feichtinger; T. Strohmer, eds.), Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Basel, 2003, pp. 31-70

[26] S. Kaliszewski; T. Omland; J. Quigg Cuntz–Li algebras from a-adic numbers, Rev. Roum. Math. Pures Appl., Volume 59 (2014) no. 3, pp. 331-370

[27] E. King; M. Skopina Quincunx multiresolution analysis for $L^{2} (Q_{2}^{2})$ , p-Adic Numbers Ultrametric Anal. Appl., Volume 2 (2010) no. 3, pp. 222-231

[28] G. Kutyniok; E. Kaniuth Zeros of the Zak transform on locally compact abelian groups, Proc. Amer. Math. Soc., Volume 126 (1998) no. 12, pp. 3561-3569

[29] N. Larsen; X. Li The 2-adic ring $C^{⁎}$ -algebra of the integers and its representations, J. Funct. Anal., Volume 262 (2012) no. 4, pp. 1392-1426

[30] F. Latrémolière; J. Packer Noncommutative solenoids and their projective modules, Commu. Noncommut. Harmon. Anal. Appl., Volume 603 (2013), pp. 35-53

[31] F. Latrémolière; J. Packer Explicit construction of equivalence bimodules between noncommutative solenoids, Contemp. Math., Amer. Math. Soc., Volume 650 (2015), pp. 111-140

[32] F. Latrémolière; J. Packer Noncommutative solenoids and the Gromov–Hausdorff propinquity, Proc. Amer. Math. Soc., Volume 145 (2017), pp. 2043-2057

[33] D. Li; H.K. Jiang Basic results of Gabor frame on local fields, Chin. Ann. Math., Volume 28 (2007) no. 2, pp. 165-176

[34] V. Losert A characterization of the minimal strongly character invariant Segal algebra, Ann. Inst. Fourier (Grenoble), Volume 30 (1980), pp. 129-139

[35] F. Luef Projections in noncommutative tori and Gabor frames, Proc. Amer. Math. Soc., Volume 139 (2011) no. 2, pp. 571-582

[36] Y.I. Lyubarskii Frames in the Bargmann space of entire functions, Entire and Subharmonic Functions, Advances in Soviet Mathematics, vol. 11, American Mathematical Society (AMS), Providence, RI, 1992, pp. 167-180

[37] Y. Manin; A. Panchishkin Introduction to Modern Number Theory, Encyclopaedia Math. Sci., vol. 49, Springer-Verlag, Berlin, 2005

[38] D. Ramakrishnan; R.J. Valenza Fourier Analysis on Number Fields, Springer, New York, 1999

[39] H. Reiter Metaplectic Groups and Segal Algebras, Lecture Notes in Mathematics, Springer, Berlin, 1989

[40] M.A. Rieffel Projective modules over higher-dimensional noncommutative tori, Can. J. Math., Volume 40 (1988) no. 2, pp. 257-338

[41] K. Seip Density theorems for sampling and interpolation in the Bargmann–Fock space. I, J. Reine Angew. Math., Volume 429 (1992), pp. 91-106

[42] F.A. Shah Gabor frames on local fields of positive characteristic, Tbil. Math. J., Volume 9 (2016) no. 2, pp. 129-139

[43] F.A. Shah Gabor-type expansions on local fields, SeMa J., Volume 75 (2018) no. 3, pp. 485-498

[44] V. Shelkovich; M. Skopina p-adic Haar multiresolution analysis and pseudo-differential operators, J. Fourier Anal. Appl., Volume 15 (2009) no. 3, pp. 366-393

Fabio Nicola Maximally localized Gabor orthonormal bases on locally compact abelian groups, Advances in Mathematics, Volume 451 (2024), p. 24 (Id/No 109786) | DOI:10.1016/j.aim.2024.109786 | Zbl:1555.43004
Ulrik Enstad; Mads S. Jakobsen; Franz Luef; Tron Omland Deformations and Balian-Low theorems for Gabor frames on the adeles, Advances in Mathematics, Volume 410 (2022), p. 46 (Id/No 108771) | DOI:10.1016/j.aim.2022.108771 | Zbl:1512.43006
Federico Bastianoni; Elena Cordero Quasi-Banach modulation spaces and localization operators on locally compact abelian groups, Banach Journal of Mathematical Analysis, Volume 16 (2022) no. 4, p. 71 (Id/No 52) | DOI:10.1007/s43037-022-00205-6 | Zbl:1506.43002
A. Austad; Eduard Ortega Groupoids and Hermitian Banach $*$ -algebras, International Journal of Mathematics, Volume 33 (2022) no. 14, p. 25 (Id/No 2250090) | DOI:10.1142/s0129167x22500902 | Zbl:1516.22004
Ulrik Enstad The density theorem for projective representations via twisted group von Neumann algebras, Journal of Mathematical Analysis and Applications, Volume 511 (2022) no. 2, p. 25 (Id/No 126072) | DOI:10.1016/j.jmaa.2022.126072 | Zbl:1494.46056
Are Austad; Franz Luef Modulation spaces as a smooth structure in noncommutative geometry, Banach Journal of Mathematical Analysis, Volume 15 (2021) no. 2, p. 31 (Id/No 35) | DOI:10.1007/s43037-020-00117-3 | Zbl:1470.46098
Ulrik Enstad The Balian-Low theorem for locally compact abelian groups and vector bundles, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 139 (2020), pp. 143-176 | DOI:10.1016/j.matpur.2019.12.005 | Zbl:1508.42034
Are Austad; Ulrik Enstad Heisenberg modules as function spaces, The Journal of Fourier Analysis and Applications, Volume 26 (2020) no. 2, p. 28 (Id/No 24) | DOI:10.1007/s00041-020-09729-7 | Zbl:1442.42068
Mads S. Jakobsen; Franz Luef Duality of Gabor frames and Heisenberg modules, Journal of Noncommutative Geometry, Volume 14 (2020) no. 4, pp. 1445-1500 | DOI:10.4171/jncg/413 | Zbl:1481.46055
Mads S. Jakobsen; Franz Luef Sampling and periodization of generators of Heisenberg modules, International Journal of Mathematics, Volume 30 (2019) no. 10, p. 37 (Id/No 1950051) | DOI:10.1142/s0129167x19500514 | Zbl:1439.46044

Cité par 10 documents. Sources : zbMATH

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