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

Authors: Karlheinz Gröchenig and Michael Leinert
Journal: J. Amer. Math. Soc. 17 (2004), 1-18
MSC (2000): Primary 22D25, 42C15; Secondary 22E25, 47B38, 47C15
DOI: https://doi.org/10.1090/S0894-0347-03-00444-2
Published electronically: September 26, 2003
MathSciNet review: 2015328
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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''.

1. 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, https://doi.org/10.1017/S0013091500018241
2. 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, https://doi.org/10.1007/BF02511155
3. Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, https://doi.org/10.1109/18.57199
4. 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, https://doi.org/10.1007/s00041-001-4018-3
5. Kenneth R. Davidson, 𝐶*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012
6. 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, https://doi.org/10.2307/2946553
7. Hans G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269–289. MR 643206, https://doi.org/10.1007/BF01320058
8. 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, https://doi.org/10.1007/BF01308667
9. 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, https://doi.org/10.1006/jfan.1996.3078
10. 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
11. Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366
12. I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., New York, 1964. MR 0205105
13. K. Gröchenig, An uncertainty principle related to the Poisson summation formula, Studia Math. 121 (1996), no. 1, 87–104. MR 1414896, https://doi.org/10.4064/sm-121-1-87-104
14. Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717
15. 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, https://doi.org/10.1090/S0273-0979-1980-14825-9
16. A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135–142. MR 323951, https://doi.org/10.1007/BF01418936
17. A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 4, 403–436. MR 1350700, https://doi.org/10.1007/s00041-001-4017-4
18. A. J. E. M. Janssen.
On rationally oversampled Weyl-Heisenberg frames.
Signal Proc., 47:239-245, 1995.
19. H. Leptin, The structure of 𝐿¹(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
20. V. Losert, On the structure of groups with polynomial growth. II, J. London Math. Soc. (2) 63 (2001), no. 3, 640–654. MR 1825980, https://doi.org/10.1017/S0024610701001983
21. Jean Ludwig, A class of symmetric and a class of Wiener group algebras, J. Functional Analysis 31 (1979), no. 2, 187–194. MR 525950, https://doi.org/10.1016/0022-1236(79)90060-0
22. M. A. Naĭmark, Normed algebras, 3rd ed., Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the second Russian edition by Leo F. Boron; Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics. MR 0438123
23. Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261
24. 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
25. 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
26. Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403–418. MR 639575, https://doi.org/10.1007/BF01465863
27. Marc A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), no. 2, 257–338. MR 941652, https://doi.org/10.4153/CJM-1988-012-9
28. Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in 𝐿₂(𝐑^{𝐝}), Duke Math. J. 89 (1997), no. 2, 237–282. MR 1460623, https://doi.org/10.1215/S0012-7094-97-08913-4
29. Thomas Strohmer, Approximation of dual Gabor frames, window decay, and wireless communications, Appl. Comput. Harmon. Anal. 11 (2001), no. 2, 243–262. MR 1848305, https://doi.org/10.1006/acha.2001.0357
30. 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, https://doi.org/10.1007/s00041-001-4011-x
31. David F. Walnut, Continuity properties of the Gabor frame operator, J. Math. Anal. Appl. 165 (1992), no. 2, 479–504. MR 1155734, https://doi.org/10.1016/0022-247X(92)90053-G
32. 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, https://doi.org/10.1006/acha.1997.0209