Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices
HTML articles powered by AMS MathViewer
- by Karlheinz Gröchenig and Michael Leinert PDF
- Trans. Amer. Math. Soc. 358 (2006), 2695-2711 Request permission
Abstract:
We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.References
- N. Atreas, J. J. Benedetto, and C. Karanikas, Local sampling for regular wavelet and Gabor expansions, Sampl. Theory Signal Image Process. 2 (2003), no. 1, 1–24. MR 2002854
- R. Balan, P. Casazza, C. Heil, and Z. Landau. Density, redundancy, and localization of frames. II. Preprint, 2005.
- Bruce A. Barnes, The spectrum of integral operators on Lebesgue spaces, J. Operator Theory 18 (1987), no. 1, 115–132. MR 912815
- Bruce A. Barnes, Symmetric Banach $\ast$-algebras: invariance of spectrum, Studia Math. 141 (2000), no. 3, 251–261. MR 1784672, DOI 10.4064/sm-141-3-251-261
- 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
- L. H. Brandenburg, On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl. 50 (1975), 489–510. MR 377523, DOI 10.1016/0022-247X(75)90006-2
- Peter G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), no. 2, 129–201. MR 1757401, DOI 10.11650/twjm/1500407227
- Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
- O. Christensen and T. Strohmer. The finite section method and problems in frame theory. Preprint, 2003.
- Elena Cordero and Karlheinz Gröchenig, Localization of frames. II, Appl. Comput. Harmon. Anal. 17 (2004), no. 1, 29–47. MR 2067914, DOI 10.1016/j.acha.2004.02.002
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Stephen Demko, William F. Moss, and Philip W. Smith, Decay rates for inverses of band matrices, Math. Comp. 43 (1984), no. 168, 491–499. MR 758197, DOI 10.1090/S0025-5718-1984-0758197-9
- Hans G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 188 (1979), no. 8-10, 451–471 (German). MR 599884
- Hans G. Feichtinger and Norbert Kaiblinger, Varying the time-frequency lattice of Gabor frames, Trans. Amer. Math. Soc. 356 (2004), no. 5, 2001–2023. MR 2031050, DOI 10.1090/S0002-9947-03-03377-4
- G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig, and C. Molitor-Braun, Weighted group algebras on groups of polynomial growth, Math. Z. 245 (2003), no. 4, 791–821. MR 2020712, DOI 10.1007/s00209-003-0571-6
- 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
- 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, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003), no. 2-4, 149–157. Frames. MR 1968117, DOI 10.1023/A:1021368609918
- Karlheinz Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004), no. 2, 105–132. MR 2054304, DOI 10.1007/s00041-004-8007-1
- 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
- Roland Hagen, Steffen Roch, and Bernd Silbermann, $C^*$-algebras and numerical analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 236, Marcel Dekker, Inc., New York, 2001. MR 1792428
- Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485, DOI 10.1137/1031129
- 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
- S. Jaffard, Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 5, 461–476 (French, with English summary). MR 1138533, DOI 10.1016/S0294-1449(16)30287-6
- 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
- Theodore W. Palmer, Spectral algebras, Rocky Mountain J. Math. 22 (1992), no. 1, 293–328. MR 1159960, DOI 10.1216/rmjm/1181072812
- Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
- Theodore W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. $*$-algebras. MR 1819503, DOI 10.1017/CBO9780511574757.003
- 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
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, École Polytech., Palaiseau, 1995, pp. Exp. No. IV, 21. MR 1362552
- Thomas Strohmer, Rates of convergence for the approximation of dual shift-invariant systems in $l^2(\mathbf Z)$, J. Fourier Anal. Appl. 5 (1999), no. 6, 599–615. MR 1752593, DOI 10.1007/BF01257194
- 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
- Thomas Strohmer, Four short stories about Toeplitz matrix calculations, Linear Algebra Appl. 343/344 (2002), 321–344. Special issue on structured and infinite systems of linear equations. MR 1878948, DOI 10.1016/S0024-3795(01)00243-9
Additional Information
- Karlheinz Gröchenig
- Affiliation: Department of Mathematics, The University of Connecticut, Storrs, Connecticut 06269-3009
- Address at time of publication: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
- Email: groch@math.uconn.edu, karlheinz.groechenig@univie.ac.at
- Michael Leinert
- Affiliation: Fakultät für Mathematik, Institut für Angewandte Mathematik, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany
- Email: leinert@math.uni-heidelberg.de
- Received by editor(s): November 4, 2003
- Received by editor(s) in revised form: August 13, 2004
- Published electronically: January 24, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2695-2711
- MSC (2000): Primary 47B37, 47A60, 46H30, 42C15
- DOI: https://doi.org/10.1090/S0002-9947-06-03841-4
- MathSciNet review: 2204052