Frames generated by compact group actions
HTML articles powered by AMS MathViewer
- by Joseph W. Iverson PDF
- Trans. Amer. Math. Soc. 370 (2018), 509-551 Request permission
Abstract:
Let $K$ be a compact group, and let $\rho$ be a representation of $K$ on a Hilbert space $\mathcal {H}_\rho$. We classify invariant subspaces of $\mathcal {H}_\rho$ in terms of range functions, and investigate frames of the form $\{\rho (\xi ) f_i\}_{\xi \in K, i \in I}$. This is done first in the setting of translation invariance, where $K$ is contained in a larger group $G$ and $\rho$ is left translation on $\mathcal {H}_\rho = L^2(G)$. For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a bracket to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences.References
- Akram Aldroubi, Carlos Cabrelli, Christopher Heil, Keri Kornelson, and Ursula Molter, Invariance of a shift-invariant space, J. Fourier Anal. Appl. 16 (2010), no. 1, 60–75. MR 2587581, DOI 10.1007/s00041-009-9068-y
- 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
- Magalí Anastasio, Carlos Cabrelli, and Victoria Paternostro, Extra invariance of shift-invariant spaces on LCA groups, J. Math. Anal. Appl. 370 (2010), no. 2, 530–537. MR 2651674, DOI 10.1016/j.jmaa.2010.05.040
- Magalí Anastasio, Carlos Cabrelli, and Victoria Paternostro, Invariance of a shift-invariant space in several variables, Complex Anal. Oper. Theory 5 (2011), no. 4, 1031–1050. MR 2861548, DOI 10.1007/s11785-010-0045-x
- Radu Victor Balan, A study of Weyl-Heisenberg and wavelet frames, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Princeton University. MR 2697492
- Davide Barbieri, Eugenio Hernández, and Azita Mayeli, Bracket map for the Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal. 37 (2014), no. 2, 218–234. MR 3223463, DOI 10.1016/j.acha.2013.12.002
- Davide Barbieri, Eugenio Hernández, and Javier Parcet, Riesz and frame systems generated by unitary actions of discrete groups, Appl. Comput. Harmon. Anal. 39 (2015), no. 3, 369–399. MR 3398942, DOI 10.1016/j.acha.2014.09.007
- Davide Barbieri, Eugenio Hernández, and Victoria Paternostro, Noncommutative shift-invariant spaces, Preprint, 2015.
- Davide Barbieri, Eugenio Hernández, and Victoria Paternostro, The Zak transform and the structure of spaces invariant by the action of an LCA group, J. Funct. Anal. 269 (2015), no. 5, 1327–1358. MR 3369940, DOI 10.1016/j.jfa.2015.06.009
- Marcin Bownik, The structure of shift-invariant subspaces of $L^2(\textbf {R}^n)$, J. Funct. Anal. 177 (2000), no. 2, 282–309. MR 1795633, DOI 10.1006/jfan.2000.3635
- Marcin Bownik, The structure of shift-modulation invariant spaces: the rational case, J. Funct. Anal. 244 (2007), no. 1, 172–219. MR 2294481, DOI 10.1016/j.jfa.2006.11.003
- 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
- Carlos Cabrelli and Victoria Paternostro, Shift-invariant spaces on LCA groups, J. Funct. Anal. 258 (2010), no. 6, 2034–2059. MR 2578463, DOI 10.1016/j.jfa.2009.11.013
- Carlos Cabrelli and Victoria Paternostro, Shift-modulation invariant spaces on LCA groups, Studia Math. 211 (2012), no. 1, 1–19. MR 2990556, DOI 10.4064/sm211-1-1
- Tuan-Yow Chien and Shayne Waldron, The projective symmetry group of a finite frame, Preprint, 2014.
- Tuan-Yow Chien and Shayne Waldron, A characterisation of projective unitary equivalence of finite frames, Preprint, 2015.
- Bradley Currey, Azita Mayeli, and Vignon Oussa, Characterization of shift-invariant spaces on a class of nilpotent Lie groups with applications, J. Fourier Anal. Appl. 20 (2014), no. 2, 384–400. MR 3200927, DOI 10.1007/s00041-013-9316-z
- Ingrid Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271–1283. MR 836025, DOI 10.1063/1.527388
- Carl de Boor, Ronald A. DeVore, and Amos Ron, The structure of finitely generated shift-invariant spaces in $L_2(\textbf {R}^d)$, J. Funct. Anal. 119 (1994), no. 1, 37–78. MR 1255273, DOI 10.1006/jfan.1994.1003
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- J. Feldman and F. P. Greenleaf, Existence of Borel transversals in groups, Pacific J. Math. 25 (1968), 455–461. MR 230837, DOI 10.2140/pjm.1968.25.455
- Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
- 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
- I. M. Gel′fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117–1120 (Russian). MR 0039154
- Frederick Greenleaf and Martin Moskowitz, Cyclic vectors for representations of locally compact groups, Math. Ann. 190 (1971), 265–288. MR 297926, DOI 10.1007/BF01431155
- A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), no. 4, 723–736. MR 747432, DOI 10.1137/0515056
- Deguang Han, Classification of finite group-frames and super-frames, Canad. Math. Bull. 50 (2007), no. 1, 85–96. MR 2296627, DOI 10.4153/CMB-2007-008-9
- Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653, DOI 10.1090/memo/0697
- Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- Henry Helson, The spectral theorem, Lecture Notes in Mathematics, vol. 1227, Springer-Verlag, Berlin, 1986. MR 873504, DOI 10.1007/BFb0101629
- Eugenio Hernández, Hrvoje Šikić, Guido Weiss, and Edward Wilson, Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, Colloq. Math. 118 (2010), no. 1, 313–332. MR 2600532, DOI 10.4064/cm118-1-17
- Eugenio Hernández, Hrvoje Šikić, Guido L. Weiss, and Edward N. Wilson, The Zak transform(s), Wavelets and multiscale analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2011, pp. 151–157. MR 2789161, DOI 10.1007/978-0-8176-8095-4_{8}
- 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
- Joseph W. Iverson, Subspaces of $L^2(G)$ invariant under translation by an abelian subgroup, J. Funct. Anal. 269 (2015), no. 3, 865–913. MR 3350733, DOI 10.1016/j.jfa.2015.03.020
- Mads Sielemann Jakobsen and Jakob Lemvig, Co-compact Gabor systems on locally compact abelian groups, J. Fourier Anal. Appl. 22 (2016), no. 1, 36–70. MR 3448915, DOI 10.1007/s00041-015-9407-0
- Gordon James and Martin Liebeck, Representations and characters of groups, 2nd ed., Cambridge University Press, New York, 2001. MR 1864147, DOI 10.1017/CBO9780511814532
- Rong Qing Jia and Charles A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II. Powers of two, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 209–246. MR 1123739
- Gerald Kaiser, A friendly guide to wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1287849
- R. A. Kamyabi Gol and R. Raisi Tousi, A range function approach to shift-invariant spaces on locally compact abelian groups, Int. J. Wavelets Multiresolut. Inf. Process. 8 (2010), no. 1, 49–59. MR 2654392, DOI 10.1142/S0219691310003365
- R. Radha and N. Shravan Kumar, Shift invariant spaces on compact groups, Bull. Sci. Math. 137 (2013), no. 4, 485–497. MR 3054272, DOI 10.1016/j.bulsci.2012.11.003
- 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
- Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of $L_2(\mathbf R^d)$, Canad. J. Math. 47 (1995), no. 5, 1051–1094. MR 1350650, DOI 10.4153/CJM-1995-056-1
- T. P. Srinivasan, Doubly invariant subspaces, Pacific J. Math. 14 (1964), 701–707. MR 164229, DOI 10.2140/pjm.1964.14.701
- Richard Vale and Shayne Waldron, Tight frames and their symmetries, Constr. Approx. 21 (2005), no. 1, 83–112. MR 2105392, DOI 10.1007/s00365-004-0560-y
- Richard Vale and Shayne Waldron, Tight frames generated by finite nonabelian groups, Numer. Algorithms 48 (2008), no. 1-3, 11–27. MR 2413275, DOI 10.1007/s11075-008-9167-x
- Richard Vale and Shayne Waldron, The construction of $G$-invariant finite tight frames, J. Fourier Anal. Appl. 22 (2016), no. 5, 1097–1120. MR 3547713, DOI 10.1007/s00041-015-9443-9
- Shayne Waldron, Group frames, Finite frames, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2013, pp. 171–191. MR 2964010, DOI 10.1007/978-0-8176-8373-3_{5}
- André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 0005741
- André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012
Additional Information
- Joseph W. Iverson
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
- Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 1109811
- Email: jiverson@math.umd.edu
- Received by editor(s): September 22, 2015
- Received by editor(s) in revised form: April 11, 2016
- Published electronically: August 15, 2017
- Additional Notes: This research was supported in part by NSF grant DMS-1265711, and by Dustin G. Mixon’s AFOSR Young Investigator Research Program award. The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 509-551
- MSC (2010): Primary 42C15, 43A77, 47A15; Secondary 22D10, 43A32
- DOI: https://doi.org/10.1090/tran/6954
- MathSciNet review: 3717988