Wiener’s lemma for twisted convolution and Gabor frames

Gröchenig, Karlheinz; Leinert, Michael

doi:10.1090/S0894-0347-03-00444-2

Wiener’s lemma for twisted convolution and Gabor frames
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by Karlheinz Gröchenig and Michael Leinert

J. Amer. Math. Soc. 17 (2004), 1-18

DOI: https://doi.org/10.1090/S0894-0347-03-00444-2

Published electronically: September 26, 2003

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Abstract:

We prove non-commutative versions of Wiener’s Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the “irrational case”.

References

Bruce A. Barnes, When is the spectrum of a convolution operator on $L^p$ independent of $p$?, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 327–332. MR 1057760, DOI 10.1017/S0013091500018241
Helmut Bölcskei and Augustus J. E. M. Janssen, Gabor frames, unimodularity, and window decay, J. Fourier Anal. Appl. 6 (2000), no. 3, 255–276. MR 1755143, DOI 10.1007/BF02511155
Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, DOI 10.1109/18.57199
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
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
George A. Elliott and David E. Evans, The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. MR 1247990, DOI 10.2307/2946553
Hans G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269–289. MR 643206, DOI 10.1007/BF01320058
Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), no. 2-3, 129–148. MR 1026614, DOI 10.1007/BF01308667
Hans G. Feichtinger and K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), no. 2, 464–495. MR 1452000, DOI 10.1006/jfan.1996.3078
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, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Chelsea Publishing Co., New York, 1964. Translated from the Russian, with a supplementary chapter. MR 0205105
K. Gröchenig, An uncertainty principle related to the Poisson summation formula, Studia Math. 121 (1996), no. 1, 87–104. MR 1414896, DOI 10.4064/sm-121-1-87-104
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
Roger Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 2, 821–843. MR 578375, DOI 10.1090/S0273-0979-1980-14825-9
A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135–142. MR 323951, DOI 10.1007/BF01418936
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

Signal Proc.

H. Leptin, The structure of $L^{1}(G)$ for locally compact groups, Operator algebras and group representations, Vol. II (Neptun, 1980) Monogr. Stud. Math., vol. 18, Pitman, Boston, MA, 1984, pp. 48–61. MR 733302
V. Losert, On the structure of groups with polynomial growth. II, J. London Math. Soc. (2) 63 (2001), no. 3, 640–654. MR 1825980, DOI 10.1017/S0024610701001983
Jean Ludwig, A class of symmetric and a class of Wiener group algebras, J. Functional Analysis 31 (1979), no. 2, 187–194. MR 525950, DOI 10.1016/0022-1236(79)90060-0
M. A. Naĭmark, Normed algebras, 3rd ed., Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the second Russian edition by Leo F. Boron. MR 0438123
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
T. Pytlik, On the spectral radius of elements in group algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 899–902 (English, with Russian summary). MR 328476
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403–418. MR 639575, DOI 10.1007/BF01465863
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
Thomas Strohmer, Approximation of dual Gabor frames, window decay, and wireless communications, Appl. Comput. Harmon. Anal. 11 (2001), no. 2, 243–262. MR 1848305, DOI 10.1006/acha.2001.0357
Richard Tolimieri and Richard S. Orr, Poisson summation, the ambiguity function, and the theory of Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 3, 233–247. MR 1353539, DOI 10.1007/s00041-001-4011-x
David F. Walnut, Continuity properties of the Gabor frame operator, J. Math. Anal. Appl. 165 (1992), no. 2, 479–504. MR 1155734, DOI 10.1016/0022-247X(92)90053-G
Meir Zibulski and Yehoshua Y. Zeevi, Analysis of multiwindow Gabor-type schemes by frame methods, Appl. Comput. Harmon. Anal. 4 (1997), no. 2, 188–221. MR 1448221, DOI 10.1006/acha.1997.0209