Publication: Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
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2013-06-20
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We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Toda-like integrable systems using the Gauss-Borel factorization of a Cantero-Moral-Velazquez moment matrix, that we construct in terms of a complex quasi-definite measure supported on the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials on the unit circle and the corresponding second kind functions. We obtain Jacobi operators, 5-term recursion relations, Christoffel-Darboux kernels, and corresponding Christoffel-Darboux formulas from this point of view in a completely algebraic way. We generalize the Cantero- Moral-Velazquez sequence of Laurent monomials, recursion relations, Christoffel-Darboux kernels, and corresponding Christoffel-Darboux formulas in this extended context. We introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and Zakharov-Shabat equations using the classical integrability theory tools. We explicitly derive the dynamical system associated with the coefficients of the orthogonal Laurent polynomials and we compare it with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szego polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, we obtain the representation of the orthogonal Laurent polynomials (and their second kind functions), using the formalism of Miwa shifts in terms of tau-functions and the subsequent bilinear equations.
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©2013 Elsevier Inc. All rights reserved.
MM thanks economical support from the Spanish Ministerio de Ciencia e Innovacion, research project FIS2008-00200 and from the Spanish Ministerio de Economia y Competitividad MTM2012-36732-C03- 01.
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[1] M.J. Ablowitz, J.F. Ladik, Nonlinear differential- difference equations, J. Math. Phys. 16 (1975) 598– 603.
[2] M.J. Ablowitz, J.F. Ladik, Nonlinear differential- difference equations and Fourier analysis, J. Math. Phys. 17 (1976) 1011–1018.
[3] M. Adler, P. van Moerbeke, Group factorization, moment matrices and Toda lattices, Int. Math. Res. Not. 12 (1997) 556–572.
[4] M. Adler, P. van Moerbeke, Generalized orthogonal polynomials, discrete KP and Riemann– Hilbert problems, Comm. Math. Phys. 207 (1999) 589–620.
[5] M. Adler, P. van Moerbeke, The spectrum of coupled random matrices, Ann. of Math. 149 (1999) 921–976.
[6] M. Adler, P. van Moerbeke, Vertex operator solutions to the discrete KP hierarchy, Comm. Math. Phys. 203 (1999) 185–210.
[7] M. Adler, P. van Moerbeke, Darboux transforms on band matrices, weights and associated polynomials, Int. Math. Res. Not. 18 (2001) 935– 984.
[8] M. Adler, P. van Moerbecke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54 (2001) 153–205.
[9] M. Adler, P. van Moerbeke, P. Vanhaecke, Moment matrices and multi-component KP, with applications to random matrix theory, Comm. Math. Phys. 286 (2009) 1–38.
[10] M. Alfaro, Una expresión de los polinomios ortogonales sobre la circunferencia unidad, Actas III J.M.H.L. 2 (1982) 1–8. (Sevilla, 1974).
[11] C. Álvarez-Fernández, U. Fidalgo, M. Mañas, The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann–Hilbert problems, Inverse Prob. 26 (2010) 055009 (p. 17).
[12] C. Älvarez-Fernández, U. Fidalgo, M. Mañas, Multiple orthogonal polynomials of mixed type: Gauss–Borel factorization and the multi-component 2D Toda hierarchy, Adv. Math. 227 (2011) 1451– 1525.
[13] R. Álvarez-Nodarse, J. Arvesú, F. Marcellán, Modifications of quasi-definite linear functionals via addition of delta and derivatives of delta Dirac functions, Indag. Math. 15 (2004) 1–20.
[14] M. Ambroladze, On exceptional sets of asymptotics relations for general orthogonal polynomials, J. Approx. Theory 82 (1995) 257–273.
[15] D. Barrios, G. López, Ratio asymptotics for orthogonal polynomials on arcs of the unit circle, Constr. Approx. 15 (1999) 1–31.
[16] M.J. Bergvelt, A.P.E. ten Kroode, Partitions, vertex operators constructions and multi- component KP equations, Pacific J. Math. 171 (1995) 23–88.
[17] E. Berriochoa, A. Cachafeiro, J. García-Amor, Connection between orthogonal polynomials on the unit circle and bounded interval, J. Comput. Appl. Math. 177 (2005) 205–223.
[18] M. Bertola, M. Gekhtman, Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions, Constr. Approx. 26 (2007) 383–430.
[19] A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njåstad, Orthogonal rational functions, in: Cambridge Monographs on Applied and Computational Mathematics, vol. 5, Cambridge University Press, Cambridge, 1999.
[20] A. Cachafeiro, F. Marcellán, C. Pérez, Lebesgue perturbation of a quasi-definite Hermitian functional. The positive definite case, Linear Algebra Appl. 369 (2003) 235–250.
[21] M. Cafasso, Matrix Biorthogonal Polynomials on the unit circle and the non-Abelian Ablowitz– Ladik hierarchy, J. Phys. A: Math. Theor. 42 (2009) 365211. [22] M.J. Cantero, L. Moral, L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003) 29–56.
[23] L. Cochran, S.C. Cooper, Orthogonal Laurent polynomials on the real line, in: S.C. Cooper, W.J. Thron (Eds.), Continued Fractions and Orthogonal Functions, in: Lecture Notes in Pure and Applied Mathematics Series, vol. 154, Marcel Dekker, NewYork, 1994, p. 47100.
[24] R. Cruz-Barroso, L. Daruis, Pablo Gonzalez-Vera, O. Njåstadb, Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle, J. Comput. Appl. Math. 200 (2007) 424–440.
[25] R. Cruz-Barroso, S. Delvaux, Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations, J. Approx. Theory 161 (2009) 65–87.
[26] R. Cruz-Barroso, P. González-Vera, A Christoffel–Darboux formula and a Favard’s theorem for Laurent orthogonal polynomials on the unit circle, J. Comput. Appl. Math. 179 (2005) 157– 173.
[27] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Operator approach to the Kadomtsev–Petviashvili equation. Transformation groups for soliton equations. III, J. Phys. Soc. Japan 50 (1981) 3806– 3812.
[28] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci. 18 (1982) 1077–1110.
[29] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations, in: M. Jimbo, T. Miwa (Eds.), Nonlinear Integrable Systems-Classical Theory and Quantum Theory, World Scientific, Singapore, 1983, pp. 39–120.
[30] C. Díaz-Mendoza, P. González-Vera, M. Jiménez-Paiz, Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Pade approximation, Math. Comp. 74 (2005) 1843–1870.
[31] L. Faybusovich, M. Gekhtman, On Schur flows, J. Phys. A: Math. Gen. 32 (1999) 4671–4680.
[32] L. Faybusovich, M. Gekhtman, Elementary Toda orbits and integrable lattices, J. Math. Phys. 41 (2000) 2905–2921.
[33] L. Faybusovich, M. Gekhtman, Inverse moment problem for elementary co-adjoint orbits, Inverse Prob. 17 (2001) 1295–1306.
[34] R. Felipe, F. Ongay, Algebraic aspects of the discrete KP hierarchy, Linear Algebra Appl. 338 (2001) 1–17.
[35] G. Freud, Orthogonal polynomials, in: Akademiai Kiadó, Budapest and Pergamon Press, Oxford, 1971, p. 1985
[36] P. García, F. Marcellan, On zeros of regular orthogonal polynomials on the unit circle, Ann. Polon. Math. 58 (1993) 287–298.
[37] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, New York, 2004.
[38] Ya.L. Geronimus, Polynomials orthogonal on a circle and their applications, Series and Approximations, Amer. Math. Soc. Transl., Serie 1, vol. 3, Amer. Math. Soc., Providence, RI (1962), pp. 1–78.
[39] E. Godoy, F. Marcellan, Orthogonal polynomials on the unit circle: distribution of zeros, J. Comput. Appl. Math. ´ 37 (1991) 195–208.
[40] L. Golinskii, Schur flows and orthogonal polynomials on the unit circle, Sb. Math. 197 (2006) 1145.
[41] G.H. Golub, C.F. Van Loan, Matrix Computations, Third ed., The John Hopkins University Press, Baltimore, CA, 1996.
[42] R. Hirota, Exact solutions of the KdV equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194.
[43] W.B. Jones, O. Njåstad, Applications of Szegő polynomials to digital signal processing, Rocky Mountain J. Math. 21 (1991) 387–436.
[44] W.B. Jones, O. Njåstad, W.J. Thron, Two-point Pade expansions for a family of analytic functions, J. Comput. Appl. Math. 9 (1983) 105–123.
[45] W.B. Jones, O. Njåstad, W.J. Thron, H. Waadeland, Szegő polynomials applied to frequency analysis, Comput. Appl. Math. 46 (1993) 217–228.
[46] W.B. Jones, W.J. Thron, Orthogonal Laurent polynomials and Gaussian quadrature, in: K.E. Gustafson, W.P. Reinhardt (Eds.), Quantum Mechanics in Mathematics Chemistry and Physics, Plenum, NewYork, 1981, pp. 449–455.
[47] W.B. Jones, W.J. Thron, H. Waadeland, A strong stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980) 503–528.
[48] R. Killip, I. Nenciu, CMV: the unitary analogue of Jacobi matrices, Comm. Pure Appl. Math. 60 (2006) 1148–1188. [49] M. Mañas, L. Martínez Alonso, C. Alvarez-Fernández, The multicomponent 2D Toda hierarchy: discrete flows and string equations, Inv. Prob. 25 (2009) 065007 (p. 31).
[50] F. Marcellán, J. Hernández, Geronimus spectral transforms and measures on the complex plane, J. Comput. Appl. Math. 219 (2008) 441–456.
[51] H.N. Mhaskar, E.B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory. 63 (1990) 30–38.
[52] A. Mukaihira, Y. Nakamura, Schur flow for orthogonal polynomials on the unit circle and its integrable discretization, J. Comput. Appl. Math. 139 (2002) 75–94.
[53] M. Mulase, Complete integrability of the Kadomtsev–Petviashvili equation, Adv. Math. 54 (1984) 57–66.
[54] I. Nenciu, Lax pairs for the Ablowitz–Ladik system via orthogonal polynomials on the unit circle, Int. Math. Res. Not. 11 (2005) 647–686.
[55] O. Njåstad, W.J. Thron, The theory of sequences of orthogonal L-polynomials, in: H. Waadeland, H. Wallin (Eds.), Pade Approximants and Continued Fractions, Det Kongelige Norske Videnskabers Selskabs Skrifter, 1983, pp. 54–91.
[56] P. Nevai, V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1–2) (1989) 99–104.
[57] K. Pan, Asymptotics for Szegő polynomials associated with Wiener–Levinson filters, J. Comput. Appl. Math. 46 (1993) 387–394.
[58] K. Pan, E.B. Saff, Asymptotics for zeros of Szegő polynomials associated with trigonometric polynomials signals, J. Approx. Theory 71 (1992) 239–251.
[59] F., M. Riesz, Uber die Randwerte einer analytischen Funktion, in: Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, 1916, pp. 27–44; W. Rudin, Real and complex analysis, in: International Series in Pure and Applied Mathematics, McGraw Hill, 1986.
[60] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, Res. Inst. Math. Sci. Kokyuroku 439 (1981) 30–46.
[61] B. Simon, Orthogonal polynomials on the unit circle, part 1: classical theory, in: AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
[62] B. Simon, Orthogonal polynomials on the unit circle, part 2: spectral theory, in: AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
[63] B. Simon, CMV matrices: five years after, J. Comput. Appl. Math. 208 (2007) 120–154.
[64] B. Simon, Zeros of OPUC and long time asymptotics of Schur and related flows, Inverse Probl. Imaging 1 (2007) 189–215.
[65] B. Simon, The Christoffel–Darboux kernel, proceedings of Symposia in Pure Mathematics 79: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday, 2008, pp. 295–336. arXiv:0806.1528.
[66] B. Simon, Szegős Theorem and its Descendants, Princeton Univertity Press, Princeton, New Jersey, 2011.
[67] G. Szegő, Orthogonal polynomials, in: American Mathematical Society Colloquium Publications, vol. XXIII., American Mathematical Society, Providence, Rhode Island, 1975.
[68] W.J. Thron, L-polynomials orthogonal on the unit circle, in: A. Cuyt (Ed.), Nonlinear Methods and Rational Approximation, Reidel Publishing Company, Dordrecht, 1988, pp. 271–278.
[69] K. Ueno, K. Takasaki, Toda lattice hierarchy, in: Group Representations and Systems of Differential Equations, Adv. Stud. Pure Math 4 (1984) 1-95.