Projections in noncommutative tori and Gabor frames

Luef, Franz

doi:10.1090/S0002-9939-2010-10489-6

Projections in noncommutative tori and Gabor frames
HTML articles powered by AMS MathViewer

by Franz Luef PDF

Proc. Amer. Math. Soc. 139 (2011), 571-582 Request permission

Abstract:

We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori and (b) the construction of tight Gabor frames for $L^2(\mathbb {R})$. The present investigation relies on interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel’s condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in noncommutative tori. In particular, the projection associated with a Gabor frame generated by a Gaussian turns out to be Boca’s projection. Our approach to Boca’s projection allows us to characterize the range of existence of Boca’s projection. The presentation of our main result provides a natural approach to the Wexler-Raz biorthogonality relations in terms of Hilbert $C^*$-modules over noncommutative tori.

References

Florin P. Boca, Projections in rotation algebras and theta functions, Comm. Math. Phys. 202 (1999), no. 2, 325–357. MR 1690050, DOI 10.1007/s002200050585
Alain Connes, $C^{\ast }$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604 (French, with English summary). MR 572645
A. Connes, An analogue of the Thom isomorphism for crossed products of a $C^{\ast }$-algebra by an action of $\textbf {R}$, Adv. in Math. 39 (1981), no. 1, 31–55. MR 605351, DOI 10.1016/0001-8708(81)90056-6
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Ingrid Daubechies, H. J. Landau, and Zeph Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl. 1 (1995), no. 4, 437–478. MR 1350701, DOI 10.1007/s00041-001-4018-3
George A. Elliott and Qing Lin, Cut-down method in the inductive limit decomposition of non-commutative tori, J. London Math. Soc. (2) 54 (1996), no. 1, 121–134. MR 1395072, DOI 10.1112/jlms/54.1.121
Hans G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269–289. MR 643206, DOI 10.1007/BF01320058
H. G. Feichtinger. Modulation spaces on locally compact Abelian groups. Technical report, January 1983. Also in R. Radha, M. Krishna, and S. Thangavelu, editors, Proc. Internat. Conf. on Wavelets and Applications, pages 1–56, Chennai, January 2002, 2003. New Delhi Allied Publishers.
H. G. Feichtinger. Modulation Spaces: Looking Back and Ahead. Sampl. Theory Signal Image Process., 5(2):109–140, 2006.
Hans G. Feichtinger and Werner Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, pp. 233–266. MR 1601091
H. G. Feichtinger and F. Luef. Wiener Amalgam Spaces for the Fundamental Identity of Gabor Analysis. Collect. Math., 57(Extra Volume (2006)):233–253, 2006.
D. Gabor. Theory of communication. J. IEEE, 93(26):429–457, 1946.
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 and Michael Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), no. 1, 1–18. MR 2015328, DOI 10.1090/S0894-0347-03-00444-2
Karlheinz Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: partial differential equations and time-frequency analysis, Fields Inst. Commun., vol. 52, Amer. Math. Soc., Providence, RI, 2007, pp. 343–366. MR 2385335
A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 4, 403–436. MR 1350700, DOI 10.1007/s00041-001-4017-4
A. J. E. M. Janssen and Thomas Strohmer, Characterization and computation of canonical tight windows for Gabor frames, J. Fourier Anal. Appl. 8 (2002), no. 1, 1–28. MR 1882363, DOI 10.1007/s00041-002-0001-x
A. J. E. M. Janssen and Thomas Strohmer, Hyperbolic secants yield Gabor frames, Appl. Comput. Harmon. Anal. 12 (2002), no. 2, 259–267. MR 1884237, DOI 10.1006/acha.2001.0376
A. J. E. M. Janssen, On generating tight Gabor frames at critical density, J. Fourier Anal. Appl. 9 (2003), no. 2, 175–214. MR 1964306, DOI 10.1007/s00041-003-0011-3
Franz Luef, On spectral invariance of non-commutative tori, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 131–146. MR 2277208, DOI 10.1090/conm/414/07805
Franz Luef, Gabor analysis, noncommutative tori and Feichtinger’s algebra, Gabor and wavelet frames, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 10, World Sci. Publ., Hackensack, NJ, 2007, pp. 77–106. MR 2428027, DOI 10.1142/9789812709080_{0}003
Franz Luef, Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces, J. Funct. Anal. 257 (2009), no. 6, 1921–1946. MR 2540994, DOI 10.1016/j.jfa.2009.06.001
Franz Luef and Yuri I. Manin, Quantum theta functions and Gabor frames for modulation spaces, Lett. Math. Phys. 88 (2009), no. 1-3, 131–161. MR 2512143, DOI 10.1007/s11005-009-0306-7
Yu. I. Lyubarskiĭ, Frames in the Bargmann space of entire functions, Entire and subharmonic functions, Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 167–180. MR 1188007
Yuri I. Manin, Theta functions, quantum tori and Heisenberg groups, Lett. Math. Phys. 56 (2001), no. 3, 295–320. EuroConférence Moshé Flato 2000, Part III (Dijon). MR 1855265, DOI 10.1023/A:1017909525397
Yu. I. Manin, Real multiplication and noncommutative geometry (ein Alterstraum), The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 685–727. MR 2077591
Yuri I. Manin, Functional equations for quantum theta functions, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 605–624. MR 2074694
Matilde Marcolli, Arithmetic noncommutative geometry, University Lecture Series, vol. 36, American Mathematical Society, Providence, RI, 2005. With a foreword by Yuri Manin. MR 2159918, DOI 10.1090/ulect/036
Kasso A. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1639–1647. MR 2051124, DOI 10.1090/S0002-9939-04-07401-5
Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572
Marc A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), no. 2, 257–338. MR 941652, DOI 10.4153/CJM-1988-012-9
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
Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
J. Wexler and S. Raz. Discrete Gabor expansions. Signal Proc., 21:207–220, 1990.