Abstract
Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play important roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved under inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.
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A. Aldroubi and K. Gröchenig. Nonuniform sampling and reconstruction in shift-invariant space, SIAM Rev, 2001, 43: 585–620.
A. Aldroubi, A. Baskakov and I. Krishtal. Slanted matrices, Banach frames, and sampling, J Funct Anal, 2008, 255: 1667–1691.
B. A. Barnes. The spectrum of integral operators on Lesbesgue spaces, J Operator Theory, 1987, 18: 115–132.
A. G. Baskakov. Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funktsional Anal i Prilozhen, 1990, 24: 64–65; translation in Funct Anal Appl, 1990, 24: 222–224.
A. G. Baskakov. Estimates for the elements of matrices of inverse operators, and harmonic analysis, Sibirsk Mat Zh, 1997, 38: 14–28.
A. G. Baskakov and I. L. Krishtal. Memory estimation of inverse operators, Arxiv: 1103.2748.
E. S. Belinskii, E. R. Liflyand, and R. M. Trigub. The Banach algebra A* and its properties, J Fourier Anal Appl, 1997, 3: 103–129.
A. Beurling. On the spectral synthesis of bounded functions, Acta Math, 1949, 81: 225–238.
B. Blackadar and J. Cuntz. Differential Banach algebra norms and smooth subalgebras of C*-algebras, J Operator Theory, 1991, 26: 255–282.
S. Bochner and R. S. Phillips. Absolutely convergent Fourier expansions for non-commutative normed rings, Ann Math, 1942, 43: 409–418.
L. Brandenburg. On idenitifying the maximal ideals in Banach Lagebras, J Math Anal Appl, 1975, 50: 489–510.
O. Christensen and T. Strohmer. The finite section method and problems in frame theory, J Approx Theory, 2005, 133: 221–237.
Q. Fang, C. E. Shin and Q. Sun. Wiener’s lemma for singular integral operators of Bessel potential type, Monatsh Math, accepted.
B. Farrell and T. Strohmer. Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group, J Operator Theory, 2010, 64: 189–205.
J. Garcia-Cuerva and J. L. Rubio de Francia. Weighted Norm Inequalities and Related Topics, Elsevier Science, New York, 1985.
I. M. Gelfand, D. A. Raikov, and G. E. Silov. Commutative Normed Rings, Chelsea, New York, 1964.
I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman. The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory, 1989, 12: 343–382.
K. Gröchenig. Wiener’s lemma: theme and variations, an introduction to spectral invariance and its applications, In: Four Short Courses on Harmonic Analysis: Wavelets, Frames, Time-Frequency Methods, and Applications to Signal and Image Analysis, editor by P. Massopust and B. Forster, Birkhauser, Boston, 2010.
K. Gröchenig and A. Klotz. Noncommutative approximation: inverse-closed subalgebras and off-diagonal decay of matrices, Constr Approx, 2010, 32: 429–466.
K. Gröchenig and A. Klotz. Norm-controlled inversion in smooth Banach algebra I, J London Math Soc, 2013, DOI: 10.1112/jlms/jdt004.
K. Gröchenig and A. Klotz. Norm-controlled inversion in smooth Banach algebra II, Math Nachr, to appear.
K. Gröchenig and M. Leinert. Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices, Trans Amer Math Soc, 2006, 358: 2695–2711.
K. Gröchenig, Z. Rzeszotnik, and T. Strohmer. Convergence analysis of the finite section method and Banach algebras of matrices, Integral Equations Operator Theory, 2010, 67: 183–202.
A. Hulanicki. On the spectrum of convolution operators on groups with polynomial growth, Invent Math, 1972, 17: 135–142.
S. Jaffard. Properiétés des matrices bien localisées prés de leur diagonale et quelques applications, Ann Inst H Poincaré, 1990, 7: 461–476.
E. Kissin and V. S. Shulman. Differential properties of some dense subalgebras of C*-algebras, Proc Edinburgh Math Soc, 1994, 37: 399–422.
I. Krishtal. Wiener’s lemma: pictures at exhibition, Rev Un Mat Argentina, 2011, 52: 61–79.
I. Krishtal, T. Strohmer and T. Wertz. Localization of matrix factorizations, Arxiv: 1305.1618.
N. Motee and Q. Sun. Sparsity measures for spatially decaying systems, in preparation.
M. A. Naimark. Normed Algebras, Wolters-Noordhoff Publishing Groningen, 1972.
N. Nikolski. In search of the invisible spectrum, Ann Inst Fourier (Grenoble), 1999, 49: 1925–1998.
M. A. Rieffel. Leibniz seminorms for “matrix algebras converge to the sphere”, In: Quanta of Maths, Clay Math Proc vol 11, Amer Math Soc, 2010, pages 543–578.
K. S. Rim, C. E. Shin and Q. Sun. Stability of localized integral operators on weighted L p spaces, Numer Funct Anal Optim, 2012, 33: 1166–1193.
I. Schur. Bemerkungen zur theorie der beschrankten bilinearformen mit unendlich vielen veranderlichen, J Reine Angew Math, 1911, 140: 1–28.
C. E. Shin and Q. Sun. Stability of localized operators, J Funct Anal, 2009, 256: 2417–2439.
J. Sjöstrand. Wiener type algebra of pseudodifferential operators, CentMath, Ecole Polytechnique, Palaiseau France, Seminaire 1994, 1995, December 1994.
E. M. Stein. Harmonic Analysis: Real—Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.
Q. Sun. Wiener’s lemma for infinite matrices with polynomial off-diagonal decay, C R Acad Sci Paris Ser I, 2005, 340: 567–570.
Q. Sun. Non-uniform sampling and reconstruction for signals with finite rate of innovations, SIAM J Math Anal, 2006, 38: 1389–1422.
Q. Sun. Wiener’s lemma for infinite matrices, Trans Amer Math Soc, 2007, 359: 3099–3123.
Q. Sun. Wiener’s lemma for localized integral operators, Appl Comput Harmon Anal, 2008, 25: 148–167.
Q. Sun. Wiener’s lemma for infinite matrices II, Constr Approx, 2011, 34: 209–235.
Q. Sun. Localized nonlinear functional equations and two sampling problems in signal processing, Adv Comput Math, 2014, DOI: 10.1007/s10444-013-9314-3.
Q. Sun and W.-S. Tang. Nonlinear frames and sparse reconstructions, submitted.
Q. Sun and J. Xian. Rate of innovation for (non)-periodic signals and optimal lower stability bound for filtering, J Fourier Anal Appl, accepted.
M. Takesaki. Theory of Operator Algebra I, Springer-Verlag, 1979.
R. Tessera. Left inverses of matrices with polynomial decay, J Funct Anal, 2010, 259: 2793–2813.
R. Tessera. The Schur algebra is not spectral in B(ℓ 2), Monatsh Math, 2010, 164: 115–118.
N. Wiener. Tauberian theorem, Ann Math, 1932, 33: 1–100.
E. Zeidler and P. R. Wadsack. Nonlinear Functional Analysis and its Applications, Vol 1, Springer-Verlag, 1998.
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The authors are partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A2005402), and National Science Foundation (DMS-1109063).
In celebration of Professor Chen Jiangong 120th anniversary
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Shin, C.E., Sun, Qy. Wiener’s lemma: localization and various approaches. Appl. Math. J. Chin. Univ. 28, 465–484 (2013). https://doi.org/10.1007/s11766-013-3215-6
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DOI: https://doi.org/10.1007/s11766-013-3215-6