Reproducing formulas for generalized translation invariant systems on locally compact abelian groups
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- by Mads Sielemann Jakobsen and Jakob Lemvig PDF
- Trans. Amer. Math. Soc. 368 (2016), 8447-8480 Request permission
Abstract:
In this paper we connect the well-established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let $\Gamma _{\!j}$, $j \in J$, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group $G$ and study systems of the form $\bigcup _{j \in J}\{ g_{j,p}(\cdot - \gamma )\}_{\gamma \in \Gamma _{\!j}, p \in P_j}$ with generators $g_{j,p}$ in $L^2(G)$ and with each $P_j$ being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical $\alpha$ local integrability condition ($\alpha$-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for $L^2(G)$. This generalizes results on generalized shift invariant systems, where each $P_j$ is assumed to be countable and each $\Gamma _{\!j}$ is a uniform lattice in $G$, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices $\Gamma _{\! j}$, our characterizations improve known results since the class of GTI systems satisfying the $\alpha$-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for $L^2(G)$. In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in $L^2(\mathbb {R}^n)$ are special cases of the same general characterizing equations.References
- S. Twareque Ali, J.-P. Antoine, and J.-P. Gazeau, Continuous frames in Hilbert space, Ann. Physics 222 (1993), no. 1, 1–37. MR 1206084, DOI 10.1006/aphy.1993.1016
- Syed Twareque Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau, Coherent states, wavelets and their generalizations, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 2000. MR 1735075, DOI 10.1007/978-1-4612-1258-4
- Marcin Bownik and Jakob Lemvig, Affine and quasi-affine frames for rational dilations, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1887–1924. MR 2746669, DOI 10.1090/S0002-9947-2010-05200-6
- Marcin Bownik and Kenneth A. Ross, The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl. 21 (2015), no. 4, 849–884. MR 3370013, DOI 10.1007/s00041-015-9390-5
- Marcin Bownik and Ziemowit Rzeszotnik, The spectral function of shift-invariant spaces on general lattices, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 49–59. MR 2066821, DOI 10.1090/conm/345/06240
- A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
- Peter G. Casazza, Ole Christensen, and A. J. E. M. Janssen, Weyl-Heisenberg frames, translation invariant systems and the Walnut representation, J. Funct. Anal. 180 (2001), no. 1, 85–147. MR 1814424, DOI 10.1006/jfan.2000.3673
- Ole Christensen, Frames and bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2008. An introductory course. MR 2428338, DOI 10.1007/978-0-8176-4678-3
- Ole Christensen and Say Song Goh, Fourier-like frames on locally compact abelian groups, J. Approx. Theory 192 (2015), 82–101. MR 3313475, DOI 10.1016/j.jat.2014.11.002
- Ole Christensen and Asghar Rahimi, Frame properties of wave packet systems in $L^2(\Bbb R^d)$, Adv. Comput. Math. 29 (2008), no. 2, 101–111. MR 2420867, DOI 10.1007/s10444-007-9038-3
- Charles K. Chui, Wojciech Czaja, Mauro Maggioni, and Guido Weiss, Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling, J. Fourier Anal. Appl. 8 (2002), no. 2, 173–200. MR 1891728, DOI 10.1007/s00041-002-0007-4
- Antonio Córdoba and Charles Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005. MR 507783, DOI 10.1080/03605307808820083
- C. Corduneanu, Almost periodic functions, Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. With the collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by Gitta Bernstein and Eugene Tomer. MR 0481915
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Filippo De Mari and Ernesto De Vito, Admissible vectors for mock metaplectic representations, Appl. Comput. Harmon. Anal. 34 (2013), no. 2, 163–200. MR 3008561, DOI 10.1016/j.acha.2012.04.001
- Hans G. Feichtinger and Thomas Strohmer (eds.), Gabor analysis and algorithms, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 1998. Theory and applications. MR 1601119, DOI 10.1007/978-1-4612-2016-9
- Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
- Massimo Fornasier and Holger Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl. 11 (2005), no. 3, 245–287. MR 2167169, DOI 10.1007/s00041-005-4053-6
- Michael Frazier, Gustavo Garrigós, Kunchuan Wang, and Guido Weiss, A characterization of functions that generate wavelet and related expansion, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), 1997, pp. 883–906. MR 1600215, DOI 10.1007/BF02656493
- Hartmut Führ, Generalized Calderón conditions and regular orbit spaces, Colloq. Math. 120 (2010), no. 1, 103–126. MR 2652610, DOI 10.4064/cm120-1-8
- Hartmut Führ, Coorbit spaces and wavelet coefficient decay over general dilation groups, Trans. Amer. Math. Soc. 367 (2015), no. 10, 7373–7401. MR 3378833, DOI 10.1090/S0002-9947-2014-06376-9
- Jean-Pierre Gabardo and Deguang Han, Frames associated with measurable spaces, Adv. Comput. Math. 18 (2003), no. 2-4, 127–147. Frames. MR 1968116, DOI 10.1023/A:1021312429186
- Gustaf Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995), no. 3, 207–226. MR 1338828, DOI 10.4064/sm-114-3-207-226
- Philipp Grohs, Continuous shearlet frames and resolution of the wavefront set, Monatsh. Math. 164 (2011), no. 4, 393–426. MR 2861594, DOI 10.1007/s00605-010-0264-2
- Philipp Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl. 17 (2011), no. 3, 506–518. MR 2803946, DOI 10.1007/s00041-010-9149-y
- Karlheinz Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, pp. 211–231. MR 1601095
- Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
- Karlheinz Gröchenig, Gitta Kutyniok, and Kristian Seip, Landau’s necessary density conditions for LCA groups, J. Funct. Anal. 255 (2008), no. 7, 1831–1850. MR 2442085, DOI 10.1016/j.jfa.2008.07.016
- Karlheinz Gröchenig and Thomas Strohmer, Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class, J. Reine Angew. Math. 613 (2007), 121–146. MR 2377132, DOI 10.1515/CRELLE.2007.094
- Eugenio Hernández, Demetrio Labate, and Guido Weiss, A unified characterization of reproducing systems generated by a finite family. II, J. Geom. Anal. 12 (2002), no. 4, 615–662. MR 1916862, DOI 10.1007/BF02930656
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- A. J. E. M. Janssen, The duality condition for Weyl-Heisenberg frames, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, pp. 33–84. MR 1601115
- Gerald Kaiser, A friendly guide to wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1287849
- Gitta Kutyniok, The local integrability condition for wavelet frames, J. Geom. Anal. 16 (2006), no. 1, 155–166. MR 2211335, DOI 10.1007/BF02930990
- Gitta Kutyniok and Demetrio Labate, The theory of reproducing systems on locally compact abelian groups, Colloq. Math. 106 (2006), no. 2, 197–220. MR 2283810, DOI 10.4064/cm106-2-3
- Gitta Kutyniok and Demetrio Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2719–2754. MR 2471937, DOI 10.1090/S0002-9947-08-04700-4
- Gitta Kutyniok and Demetrio Labate (eds.), Shearlets, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2012. Multiscale analysis for multivariate data. MR 2896273, DOI 10.1007/978-0-8176-8316-0
- Demetrio Labate, Guido Weiss, and Edward Wilson, An approach to the study of wave packet systems, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 215–235. MR 2066831, DOI 10.1090/conm/345/06250
- R. S. Laugesen, N. Weaver, G. L. Weiss, and E. N. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (2002), no. 1, 89–102. MR 1881293, DOI 10.1007/BF02930862
- Shidong Li, Yulong Liu, and Tiebin Mi, Sparse dual frames and dual Gabor functions of minimal time and frequency supports, J. Fourier Anal. Appl. 19 (2013), no. 1, 48–76. MR 3019770, DOI 10.1007/s00041-012-9243-4
- Jerry Lopez and Deguang Han, Discrete Gabor frames in $\ell ^2(\Bbb {Z}^d)$, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3839–3851. MR 3091773, DOI 10.1090/S0002-9939-2013-11875-7
- A. Rahimi, A. Najati, and Y. N. Dehghan, Continuous frames in Hilbert spaces, Methods Funct. Anal. Topology 12 (2006), no. 2, 170–182. MR 2238038
- Hans Reiter and Jan D. Stegeman, Classical harmonic analysis and locally compact groups, 2nd ed., London Mathematical Society Monographs. New Series, vol. 22, The Clarendon Press, Oxford University Press, New York, 2000. MR 1802924
- Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbf R^d)$, Duke Math. J. 89 (1997), no. 2, 237–282. MR 1460623, DOI 10.1215/S0012-7094-97-08913-4
- Amos Ron and Zuowei Shen, Generalized shift-invariant systems, Constr. Approx. 22 (2005), no. 1, 1–45. MR 2132766, DOI 10.1007/s00365-004-0563-8
- Wenchang Sun, $G$-frames and $g$-Riesz bases, J. Math. Anal. Appl. 322 (2006), no. 1, 437–452. MR 2239250, DOI 10.1016/j.jmaa.2005.09.039
- Xihua Wang, The study of wavelets from the properties of their Fourier transforms, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–Washington University in St. Louis. MR 2692675
- J. Yao, P. Krolak, and C. Steele. The generalized Gabor transform. IEEE Transactions on image processing, 4(7):978–988, 1995.
Additional Information
- Mads Sielemann Jakobsen
- Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800 Kgs. Lyngby, Denmark
- Email: msja@dtu.dk
- Jakob Lemvig
- Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800 Kgs. Lyngby, Denmark
- Email: jakle@dtu.dk
- Received by editor(s): May 19, 2014
- Received by editor(s) in revised form: October 8, 2014
- Published electronically: February 10, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8447-8480
- MSC (2010): Primary 42C15, 43A32, 43A70; Secondary 43A60, 46C05
- DOI: https://doi.org/10.1090/tran/6594
- MathSciNet review: 3551577