Noncommutative Fourier Analysis on Invariant Subspaces: Frames of Unitary Orbits and Hilbert Modules over Group von Neumann Algebras
DOI:
https://doi.org/10.6092/issn.2240-2829/6689Keywords:
Frames, Group representations, Hilbert modules, Fourier analysisAbstract
This is a joint work with E. Hernández, J. Parcet and V. Paternostro. We will discuss the structure of bases and frames of unitary orbits of discrete groups in invariant subspaces of separable Hilbert spaces. These invariant spaces can be characterized, by means of Fourier intertwining operators, as modules whose rings of coefficients are given by the group von Neumann algebra, endowed with an unbounded operator valued pairing which defines a noncommutative Hilbert structure. Frames and bases obtained by countable families of orbits have noncommutative counterparts in these Hilbert modules, given by countable families of operators satisfying generalized reproducing conditions. These results extend key notions of Fourier and wavelet analysis to general unitary actions of discrete groups, such as crystallographic transformations on the Euclidean plane or discrete Heisenberg groups.
References
D. Barbieri, E. Hernández, A. Mayeli, Bracket map for the Heisenberg group and the characterization of cyclic subspaces. Appl. Comput. Harmon. Anal. 37:218-234 (2014).
D. Barbieri, E. Hernández, J. Parcet, Riesz and frame systems generated by unitary actions of discrete groups. Appl. Comput. Harmon. Anal. 39:369-399 (2015).
D. Barbieri, E. Hernández, V. Paternostro, The Zak transform and the structure of spaces invariant by the action of an LCA group. J. Funct. Anal. 269:1327-1358 (2015).
D. Barbieri, E. Hernández, V. Paternostro, Noncommutative shift-invariant spaces. Preprint, http://arxiv.org/abs/1506.08942
J. J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5:389-427 (1998).
M. Bownik, The structure of shift-invariant subspaces of L2(Rn). J. Funct. Anal. 177 (2):282-309 (2000).
M. Bownik, K. A. Ross, The structure of translation-invariant spaces on locally compact abelian groups. J. Fourier Anal. Appl. 21:849-884 (2015).
C. Cabrelli, V. Paternostro, Shift-invariant spaces on LCA groups. J. Funct. Anal. 258:2034-2059 (2010).
P. G. Casazza, The art of frame theory. Taiwanese J. Math. 4:129-201 (2000).
P. G. Casazza, J. C. Tremain, The Kadison-Singer problem in Mathematics and Engineering. Proc. Natl. Acad. Sci. USA 103:2032-2039 (2006).
A. Connes, Noncommutative geometry. Academic Press 1994.
J. B. Conway, A course in functional analysis. Springer, 2nd ed. 1990.
M. Enock, J. M. Schwartz, Kac algebras and duality of locally compact groups. Springer 1992.
P. Eymard, L'algébre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92:181-236 (1964).
M. Frank, D. R. Larson, Frames in Hilbert C*-modules and C*-algebras. J. Operator Theory 48:273-
(2002).
E. Hernández, H. Šikić, G. Weiss, E. Wilson, Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform. Colloq. Math. 118:313-332 (2010).
E. Hernández, G. Weiss, A first course on wavelets. CRC Press 1996.
M. Junge, D. Sherman, Noncommutative Lp modules. J. Operator Theory 53:3-34 (2005).
R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol.1 and Vol. 2. Academic Press 1983.
R. A. Kunze, Lp Fourier transforms on locally compact unimodular groups. Trans. Am. Math. Soc. 89(2):519-540 (1958).
F. Luef and Y. I. Manin. Quantum theta functions and Gabor frames for modulation spaces. Lett. Math. Phys., 88(1-3):131161, 2009.
S. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation IEEE T. Pattern Anal. 31:674-693 (1989).
Y. Meyer, R. Coifman, Wavelets. Calderón-Zygmund and multilinear operators. Cambridge University Press 1997.
A. Ron, Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd). Canad. J. Math. 47:1051-1094 (1995).
E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press 1971.
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