Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

Multiple orthogonality is considered in the realm of a Gauss--Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss--Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel--Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss--Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the $\tau$-function representation of the multiple orthogonality is given.Comment: 53 pages. In this version minor revisions regarding the Christoffel-Darboux operators are performe

General results on the convergence of multipoint Hermite-Padé approximants of Nikishin systems

19 pages, no figures.-- MSC1991 codes: Primary 30E10, 42C05.MR#: MR2263738 (2007g:42041)Zbl#: Zbl 1105.30024We consider simultaneous approximation of Nikishin systems of functions by means of rational vector functions which are constructed interpolating along a prescribed table of points. We give general conditions for the uniform convergence of such approximants with a geometric rate under very weak assumptions.The work of both authors was supported by Ministerio de Ciencia y Tecnología under grant BFM 2003-06335-C03-02. The second author was also partially supported by NATO PST.CLG.979738 and INTAS 03-51-6637.Publicad

Generalized Hermite-Padé approximation for Nikishin systems of three functions

9 pages, no figures.-- MSC1991 code: Primary 42C05.-- Issue title: "Special Functions, Information Theory, and Mathematical Physics". Special issue dedicated to Professor Jesús Sánchez Dehesa on the occasion of his 60th birthday.Zbl#: Zbl pre05650072Nikishin systems of three functions are considered. For such systems, the rate of convergence of simultaneous interpolating rational approximations with partially prescribed poles is studied. The solution is described in terms of the solution of a vector equilibrium problem in the presence of a vector external field.First author’s research supported by grants MTM 2006-13000-C03-02 from Ministerio de Ciencia y Tecnología and CCG 06–UC3M/ESP–0690 of Universidad Carlos III de Madrid-Comunidad de Madrid and by grant SFRH/BPD/31724/2006 from Fundação para a Ciência e a Tecnologia. Second author’s research supported by grants MTM 2006-13000-C03-02 from Ministerio de Ciencia y Tecnología and CCG 06–UC3M/ESP–0690 of Universidad Carlos III de Madrid-Comunidad de Madrid.Publicad

Rate of convergence of generalized Hermite-Padé approximants of Nikishin systems

32 pages, no figures.-- MSC1991 code: Primary, 42CD5.MR#: MR2186304 (2006h:41017)Zbl#: Zbl 1136.42306We study the rate of convergence of interpolating simultaneous rational approximations with partially prescribed poles to so-called Nikishin systems of functions. To this end, a vector equilibrium problem in the presence of a vector external field is solved which is used to describe the asymptotic behavior of the corresponding second-type functions which appear.The work of both authors was supported by the Ministerio de Ciencia y Tecnología under grant BFM 2003-06335-C03-02. The second author was also partially supported by NATO PST.CLG.979738 and INTAS 03-51-6637.Publicad

On perfect Nikishin systems

12 pages, no figures.-- MSC1991 code: Primary 42C05.-- Publisher's full-text version available Open Access at: http://www.heldermann-verlag.de/cmf/cmf02/cmf0224.pdfMR#: MR2038130 (2005c:42026)Zbl#: Zbl 1065.42020We prove perfectness for Nikishin systems made up of three functions and apply this to the convergence of the associated Hermite-Padé approximant.The work of both authors was partially supported by Dirección General de Enseñanza Superior under grant BFM2000-0206-C04-01 and the second author by grants PRAXIS XXI BCC-22201/99 and INTAS 00-272.Publicad

An extension of Markov's Theorem

We give a general sufficient condition for the uniform convergence of sequences of type II Hermite-Padé approximants associated with Nikishin systems of functions

The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann-Hilbert problems

15 pages.-- ArXiv pre-print available at: http://arxiv.org/abs/0911.0941Submitted to: Inverse ProblemsWe consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.MM thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200 and UF thanks economical support from the Spanish Ministerio de Ciencia e Innovación research projects MTM2006-13000-C03-02 and MTM2007-62945 and from Comunidad de Madrid/Universidad Carlos III de Madrid project CCG07-UC3M/ESP-3339.No publicad

Convergence and computation of simultaneous rational quadrature formulas

22 pages, no figures.-- MSC2000 codes: Primary 41A55. Secondary 41A28, 65D32.MR#: MR2286008 (2008a:65049)Zbl#: Zbl 1168.65326We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.The work of U.F.P. and G.L.L. was partially supported by Dirección General de Enseñanza Superior under grant BFM2003-06335-C03-02 and of G.L.L. by INTAS under Grant INTAS 03-51-6637. The work of J.R.I. was supported by a research grant from the Ministerio de Educación y Ciencia, project code MTM 2005-01320.Publicad

Hermite-Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights