On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients

Sakhnovich, Alexander

doi:10.1090/tran/7770

On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients
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Trans. Amer. Math. Soc. 372 (2019), 5547-5570 Request permission

Abstract:

The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. We consider the 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal information, which is necessary to recover the inverse matrices, and give a complete characterization of the inverse matrices. A 2-D analogue of the important Ambartsumyan and Sobolev formulas for the corresponding reflection coefficients is derived as well.

References

D. Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, and A. L. Sakhnovich, Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials, Integral Equations Operator Theory 68 (2010), no. 1, 115–150. MR 2677891, DOI 10.1007/s00020-010-1803-x
V. A. Ambartsumyan, Scientific works. Vol. I, Academy of Sciences of Armenian SSR, Erevan, 1960 (Russian).
Dario Bini and Victor Y. Pan, Polynomial and matrix computations. Vol. 1, Progress in Theoretical Computer Science, Birkhäuser Boston, Inc., Boston, MA, 1994. Fundamental algorithms. MR 1289412, DOI 10.1007/978-1-4612-0265-3
Dario Bini, Sarah Dendievel, Guy Latouche, and Beatrice Meini, General solution of the Poisson equation for quasi-birth-and-death processes, SIAM J. Appl. Math. 76 (2016), no. 6, 2397–2417. MR 3585031, DOI 10.1137/16M1065045
J. M. Bogoya, A. Böttcher, S. M. Grudsky, and E. A. Maximenko, Eigenvectors of Hermitian Toeplitz matrices with smooth simple-loop symbols, Linear Algebra Appl. 493 (2016), 606–637. MR 3452760, DOI 10.1016/j.laa.2015.12.017
Albrecht Böttcher and Sergei M. Grudsky, Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. MR 2179973, DOI 10.1137/1.9780898717853
Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Prepared jointly with Alexei Karlovich. MR 2223704
Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
Yong Chen, Kei Ji Izuchi, and Young Joo Lee, Kernels of Toeplitz operators on the Hardy space over the bidisk, J. Funct. Anal. 272 (2017), no. 9, 3869–3903. MR 3620714, DOI 10.1016/j.jfa.2017.01.002
E. Chevallier, T. Forget, F. Barbaresco, and J. Angulo, Kernel density estimation on the Siegel space with an application to radar processing, Entropy 18 (2016), 396.
Darren G. Crowdy and Elena Luca, Solving Wiener-Hopf problems without kernel factorization, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2170, 20140304, 20. MR 3245450, DOI 10.1098/rspa.2014.0304
A. M. Davie and N. P. Jewell, Toeplitz operators in several complex variables, J. Functional Analysis 26 (1977), no. 4, 356–368. MR 0461195, DOI 10.1016/0022-1236(77)90020-9
Maxim Derevyagin and Brian Simanek, On Szegő’s theorem for a nonclassical case, J. Funct. Anal. 272 (2017), no. 6, 2487–2503. MR 3603306, DOI 10.1016/j.jfa.2016.07.009
Allen Devinatz and Marvin Shinbrot, General Wiener-Hopf operators, Trans. Amer. Math. Soc. 145 (1969), 467–494. MR 251573, DOI 10.1090/S0002-9947-1969-0251573-0
B. Fritzsche, B. Kirstein, I. Ya. Roitberg, and A. L. Sakhnovich, Operator identities corresponding to inverse problems for Dirac systems, Indag. Math. (N.S.) 23 (2012), no. 4, 690–700. MR 2991918, DOI 10.1016/j.indag.2012.05.002
F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601, DOI 10.1090/chel/345
Y. Genin and Y. G. Kamp, Two-dimensional stability and orthogonal polynomials on the hypercircle, Proc. IEEE 65 (1977), 873–881.
I. Gohberg (ed.), Continuous and discrete Fourier transforms, extension problems and Wiener-Hopf equations, Operator Theory: Advances and Applications, vol. 58, Birkhäuser Verlag, Basel, 1992. MR 1183740, DOI 10.1007/978-3-0348-8596-6
I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on a half line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. (2) 14 (1960), 217–287. MR 0113114, DOI 10.1090/trans2/014/09
I. C. Gohberg and A. A. Semencul, The inversion of finite Toeplitz matrices and their continual analogues, Mat. Issled. 7 (1972), no. 2(24), 201–223, 290 (Russian). MR 0353038
Vadim Gorin, The $q$-Gelfand-Tsetlin graph, Gibbs measures and $q$-Toeplitz matrices, Adv. Math. 229 (2012), no. 1, 201–266. MR 2854175, DOI 10.1016/j.aim.2011.08.016
Ulf Grenander and Gábor Szegő, Toeplitz forms and their applications, 2nd ed., Chelsea Publishing Co., New York, 1984. MR 890515
Georg Heinig and Karla Rost, Algebraic methods for Toeplitz-like matrices and operators, Mathematical Research, vol. 19, Akademie-Verlag, Berlin, 1984. MR 756628, DOI 10.1007/978-3-0348-6241-7
B. Jeuris and R. Vandebril, The Kähler mean of block-Toeplitz matrices with Toeplitz structured blocks, SIAM J. Matrix Anal. Appl. 37 (2016), no. 3, 1151–1175. MR 3543154, DOI 10.1137/15M102112X
J. H. Justice, A Levinson type algorithm for two dimensional Wiener filtering using bivariate Szegő polynomials, Proc. IEEE 65 (1977), 882–886.
M. A. Kaashoek and F. van Schagen, On inversion of certain structured linear transformations related to block Toeplitz matrices, A panorama of modern operator theory and related topics, Oper. Theory Adv. Appl., vol. 218, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 377–386. MR 2931937, DOI 10.1007/978-3-0348-0221-5_{1}6
Sergey I. Kabanikhin, Nikita S. Novikov, Ivan V. Oseledets, and Maxim A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, J. Inverse Ill-Posed Probl. 23 (2015), no. 6, 687–700. MR 3430922, DOI 10.1515/jiip-2015-0083
M. Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21 (1954), 501–509. MR 62867
Thomas Kailath, Sun Yuan Kung, and Martin Morf, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl. 68 (1979), no. 2, 395–407. MR 533501, DOI 10.1016/0022-247X(79)90124-0
T. Kailath, S. Y. Kung, and M. Morf, Displacement ranks of a matrix, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 5, 769–773. MR 537629, DOI 10.1090/S0273-0979-1979-14659-7
Thomas Kailath and Ali H. Sayed, Displacement structure: theory and applications, SIAM Rev. 37 (1995), no. 3, 297–386. MR 1355506, DOI 10.1137/1037082
T. Kailath and A. H. Sayed (eds.), Fast reliable algorithms for matrices with structure, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. MR 1715813, DOI 10.1137/1.9781611971354
N. Kalouptsidis, G. Carayannis, and D. Manolakis, On block matrices with elements of special structure, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP-82) 7, IEEE, New York, 1982, pp. 1744–1747.
M. G. Krein, Integral equations on the half-line with a kernel depending on the difference of the arguments, Amer. Math. Soc. Transl. 22 (1962), 163–288.
Norman Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys. Mass. Inst. Tech. 25 (1947), 261–278. MR 19257, DOI 10.1002/sapm1946251261
B. Levy, $2$-D polynomial and rational matrices and their application for the modelling of $2$-D dynamical systems, 1981. Thesis (Ph.D.)–Stanford University.
D. I. Nagirner,Viktor V. Sobolev and radiative transfer theory, J. Quant. Spectrosc. Radiat. Transfer 183 (2016), 4–37.
V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, “Nauka”, Moscow, 1982 (Russian). MR 678923
Frank Nielsen and Rajendra Bhatia (eds.), Matrix information geometry, Springer, Heidelberg, 2013. MR 2964195, DOI 10.1007/978-3-642-30232-9
B. Noble, Methods based on the Wiener–Hopf technique, Pergamon Press, London, 1958.
Vadim Olshevsky, Ivan Oseledets, and Eugene Tyrtyshnikov, Tensor properties of multilevel Toeplitz and related matrices, Linear Algebra Appl. 412 (2006), no. 1, 1–21. MR 2180855, DOI 10.1016/j.laa.2005.03.040
Victor Y. Pan, John Svadlenka, and Liang Zhao, Estimating the norms of random circulant and Toeplitz matrices and their inverses, Linear Algebra Appl. 468 (2015), 197–210. MR 3293250, DOI 10.1016/j.laa.2014.06.027
J. Pestana and A. J. Wathen, A preconditioned MINRES method for nonsymmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl. 36 (2015), no. 1, 273–288. MR 3323542, DOI 10.1137/140974213
Alexei Poltoratski, Toeplitz approach to problems of the uncertainty principle, CBMS Regional Conference Series in Mathematics, vol. 121, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2015. MR 3309830, DOI 10.1090/cbms/121
Philippe Rambour and Abdellatif Seghier, Asymptotic inversion of Toeplitz matrices with one singularity in the symbol, C. R. Math. Acad. Sci. Paris 347 (2009), no. 9-10, 489–494 (English, with English and French summaries). MR 2576895, DOI 10.1016/j.crma.2009.02.024
A. L. Sahnovič, A certain method of inverting Toeplitz matrices, Mat. Issled. 8 (1973), no. 4(30), 180–186, 199. MR 0346581
A. L. Sahnovič, Continuation of block Toeplitz matrices, Functional analysis, No. 14 (Russian), Ul′yanovsk. Gos. Ped. Inst., Ul′yanovsk, 1980, pp. 116–127 (Russian). MR 596640
Alexander Sakhnovich, Toeplitz matrices with an exponential growth of entries and the first Szegö limit theorem, J. Funct. Anal. 171 (2000), no. 2, 449–482. MR 1745627, DOI 10.1006/jfan.1999.3543
Alexander L. Sakhnovich, Lev A. Sakhnovich, and Inna Ya. Roitberg, Inverse problems and nonlinear evolution equations, De Gruyter Studies in Mathematics, vol. 47, De Gruyter, Berlin, 2013. Solutions, Darboux matrices and Weyl-Titchmarsh functions. MR 3098432, DOI 10.1515/9783110258615
A. L. Sakhnovich and I. M. Spitkovskiĭ, Block-Toeplitz matrices and associated properties of a Gaussian model on the half axis, Teoret. Mat. Fiz. 63 (1985), no. 1, 154–160 (Russian, with English summary). MR 794478
L. A. Sahnovič, Operators, similar to unitary operators, with absolutely continuous spectrum, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 51–63 (Russian). MR 0232243
L. A. Sahnovič, An integral equation with a kernel dependent on the difference of the arguments, Mat. Issled. 8 (1973), no. 2(28), 138–146, 193 (Russian). MR 0331129
L. A. Sakhnovich, Problems of factorization and operator identities, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 4–55, 240 (Russian). MR 832409
L. A. Sakhnovich, Extremal trigonometric and power polynomials of several variables, Funktsional. Anal. i Prilozhen. 38 (2004), no. 1, 88–91 (Russian); English transl., Funct. Anal. Appl. 38 (2004), no. 1, 72–74. MR 2061796, DOI 10.1023/B:FAIA.0000024872.39055.62
Lev A. Sakhnovich, Integral equations with difference kernels on finite intervals, Operator Theory: Advances and Applications, vol. 84, Birkhäuser/Springer, Cham, 2015. Second edition, revised and extended. MR 3287271, DOI 10.1007/978-3-319-16489-2
Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1289670
Donald Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491–526. MR 2363975, DOI 10.7153/oam-01-29
David Sharpe, Rings and factorization, Cambridge University Press, Cambridge, 1987. MR 905677, DOI 10.1017/CBO9780511565960
Marvin Shinbrot, A class of difference kernels, Proc. Amer. Math. Soc. 13 (1962), 399–406. MR 137966, DOI 10.1090/S0002-9939-1962-0137966-X
Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
V. V. Sobolev, Light scattering in planetary atmospheres, International Series of Monographs in Natural Philosophy, vol. 76, Pergamon Press, Oxford, 1975.
William F. Trench, Weighting coefficients for the prediction of stationary time series from the finite past, SIAM J. Appl. Math. 15 (1967), 1502–1510. MR 225458, DOI 10.1137/0115132
Mati Wax and Thomas Kailath, Efficient inversion of Toeplitz-block Toeplitz matrix, IEEE Trans. Acoust. Speech Signal Process. 31 (1983), no. 5, 1218–1221. MR 726463, DOI 10.1109/TASSP.1983.1164208
Harold Widom, Wiener-Hopf integral equations, The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 60, Amer. Math. Soc., Providence, RI, 1997, pp. 391–405. MR 1460293, DOI 10.1090/pspum/060/1460293
Pengpeng Xie and Yimin Wei, The stability of formulae of the Gohberg-Semencul-Trench type for Moore-Penrose and group inverses of Toeplitz matrices, Linear Algebra Appl. 498 (2016), 117–135. MR 3478554, DOI 10.1016/j.laa.2015.01.029