Wiener’s Lemma for infinite matrices. (English) Zbl 1131.47013
Summary: The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices \( {\mathcal W}:=\{(a(j-j'))_{j,j'\in\mathbb{Z}^d}: \sum_{j\in \mathbb{Z}^d} | a(j)|<\infty\}\). In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those non-commutative algebras to have a similar property to Wiener’s lemma for the commutative algebra \( {\mathcal W}\). In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras.
MSC:
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 41A15 | Spline approximation |
References:
| [1] | Akram Aldroubi and Karlheinz Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev. 43 (2001), no. 4, 585 – 620. · Zbl 0995.42022 · doi:10.1137/S0036144501386986 |
| [2] | 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. · Zbl 1049.42018 |
| [3] | Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Density, overcompleteness, and localization of frames. I. Theory, J. Fourier Anal. Appl. 12 (2006), no. 2, 105 – 143. , https://doi.org/10.1007/s00041-006-6022-0 Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau, Density, overcompleteness, and localization of frames. II. Gabor systems, J. Fourier Anal. Appl. 12 (2006), no. 3, 309 – 344. · Zbl 1097.42022 · doi:10.1007/s00041-005-5035-4 |
| [4] | Bruce A. Barnes, The spectrum of integral operators on Lebesgue spaces, J. Operator Theory 18 (1987), no. 1, 115 – 132. · Zbl 0646.47033 |
| [5] | A. G. Baskakov, Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 64 – 65 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 3, 222 – 224 (1991). · Zbl 0728.47021 · doi:10.1007/BF01077964 |
| [6] | A. G. Baskakov, Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis, Sibirsk. Mat. Zh. 38 (1997), no. 1, 14 – 28, i (Russian, with Russian summary); English transl., Siberian Math. J. 38 (1997), no. 1, 10 – 22. · Zbl 0870.43003 · doi:10.1007/BF02674895 |
| [7] | L. H. Brandenburg, On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl. 50 (1975), 489 – 510. · Zbl 0302.46042 · doi:10.1016/0022-247X(75)90006-2 |
| [8] | Ole Christensen and Thomas Strohmer, The finite section method and problems in frame theory, J. Approx. Theory 133 (2005), no. 2, 221 – 237. · Zbl 1078.42024 · doi:10.1016/j.jat.2005.01.001 |
| [9] | Charles K. Chui, Wenjie He, and Joachim Stöckler, Nonstationary tight wavelet frames. II. Unbounded intervals, Appl. Comput. Harmon. Anal. 18 (2005), no. 1, 25 – 66. · Zbl 1067.42022 · doi:10.1016/j.acha.2004.09.004 |
| [10] | Albert Cohen, Ingrid Daubechies, and Pierre Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 54 – 81. · Zbl 0795.42018 · doi:10.1006/acha.1993.1005 |
| [11] | Albert Cohen and Nira Dyn, Nonstationary subdivision schemes and multiresolution analysis, SIAM J. Math. Anal. 27 (1996), no. 6, 1745 – 1769. · Zbl 0862.41013 · doi:10.1137/S003614109427429X |
| [12] | Ronald R. Coifman and Mauro Maggioni, Diffusion wavelets, Appl. Comput. Harmon. Anal. 21 (2006), no. 1, 53 – 94. · Zbl 1095.94007 · doi:10.1016/j.acha.2006.04.004 |
| [13] | Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. · Zbl 0224.43006 |
| [14] | Elena Cordero and Karlheinz Gröchenig, Localization of frames. II, Appl. Comput. Harmon. Anal. 17 (2004), no. 1, 29 – 47. · Zbl 1061.42016 · doi:10.1016/j.acha.2004.02.002 |
| [15] | Carl de Boor, A bound on the \?_{\infty }-norm of \?\(_{2}\)-approximation by splines in terms of a global mesh ratio, Math. Comp. 30 (1976), no. 136, 765 – 771. · Zbl 0345.65004 |
| [16] | Stephen Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal. 14 (1977), no. 4, 616 – 619. · Zbl 0367.65024 · doi:10.1137/0714041 |
| [17] | Gero Fendler, Karlheinz Gröchenig, and Michael Leinert, Symmetry of weighted \?\textonesuperior -algebras and the GRS-condition, Bull. London Math. Soc. 38 (2006), no. 4, 625 – 635. · Zbl 1096.43002 · doi:10.1112/S0024609306018777 |
| [18] | Massimo Fornasier and Karlheinz Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (2005), no. 3, 395 – 415. · Zbl 1130.41304 · doi:10.1007/s00365-004-0592-3 |
| [19] | Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. · Zbl 0966.42020 |
| [20] | Karlheinz Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003), no. 2-4, 149 – 157. Frames. · Zbl 1019.42019 · doi:10.1023/A:1021368609918 |
| [21] | Karlheinz Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004), no. 2, 105 – 132. · Zbl 1055.42018 · doi:10.1007/s00041-004-8007-1 |
| [22] | K. Gröchenig, Time-frequency analysis of Sjöstrand’s class, Rev. Mat. Iberoam., 22(2006), 703-724. · Zbl 1127.35089 |
| [23] | Karlheinz Gröchenig and Michael Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), no. 1, 1 – 18. · Zbl 1037.22012 |
| [24] | Karlheinz Gröchenig and Michael Leinert, Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2695 – 2711. · Zbl 1105.46032 |
| [25] | Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, Springer-Verlag, New York-Berlin, 1979. · Zbl 0439.43001 |
| [26] | S. Jaffard, Propriétés des matrices ”bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 5, 461 – 476 (French, with English summary). · Zbl 0722.15004 |
| [27] | Rong Qing Jia and Charles A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II. Powers of two, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 209 – 246. · Zbl 0777.41013 |
| [28] | Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257 – 270. , https://doi.org/10.1016/0001-8708(79)90012-4 Roberto A. Macías and Carlos Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 271 – 309. · Zbl 0431.46019 · doi:10.1016/0001-8708(79)90013-6 |
| [29] | Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257 – 270. , https://doi.org/10.1016/0001-8708(79)90012-4 Roberto A. Macías and Carlos Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 271 – 309. · Zbl 0431.46019 · doi:10.1016/0001-8708(79)90013-6 |
| [30] | D. J. Newman, A simple proof of Wiener’s 1/\? theorem, Proc. Amer. Math. Soc. 48 (1975), 264 – 265. · Zbl 0296.42017 |
| [31] | Gerlind Plonka, Periodic spline interpolation with shifted nodes, J. Approx. Theory 76 (1994), no. 1, 1 – 20. · Zbl 0803.41003 · doi:10.1006/jath.1994.1001 |
| [32] | Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. · Zbl 0732.47001 |
| [33] | 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. · Zbl 0880.35145 |
| [34] | Thomas Strohmer, Rates of convergence for the approximation of dual shift-invariant systems in \?²(\?), J. Fourier Anal. Appl. 5 (1999), no. 6, 599 – 615. · Zbl 0981.42020 · doi:10.1007/BF01257194 |
| [35] | 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. · Zbl 0999.65026 · doi:10.1016/S0024-3795(01)00243-9 |
| [36] | Qiyu Sun, Wiener’s lemma for infinite matrices with polynomial off-diagonal decay, C. R. Math. Acad. Sci. Paris 340 (2005), no. 8, 567 – 570 (English, with English and French summaries). · Zbl 1069.42018 · doi:10.1016/j.crma.2005.03.002 |
| [37] | Q. Sun, Frames in spaces with finite rate of innovations, Adv. Comput. Math., 27(2007), To appear. |
| [38] | Q. Sun, Non-uniform sampling and reconstruction for signals with finite rate of innovations, SIAM J. Math. Anal., To appear. |
| [39] | Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1 – 100. · Zbl 0004.05905 · doi:10.2307/1968102 |
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