134 research outputs found
Nikishin systems are perfect
K. Mahler introduced the concept of perfect systems in the general theory he
developed for the simultaneous Hermite-Pade approximation of analytic
functions. We prove that Nikishin systems are perfect providing, by far, the
largest class of systems of functions for which this important property holds.
As consequences, in the context of Nikishin systems, we obtain: an extension of
Markov's theorem to simultaneous Hermite-Pade approximation, a general result
on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the
logarithmic asymptotics of general sequences of multiple orthogonal
polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio
asymptotics of mixed type multiple orthogonal polynomials.Comment: 39 page
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 -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 TecnologĂa 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 JesĂșs SĂĄnchez 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 authorâs research supported by grants MTM 2006-13000-C03-02 from Ministerio de Ciencia y TecnologĂa and CCG 06âUC3M/ESPâ0690 of Universidad Carlos III de Madrid-Comunidad de Madrid and by grant SFRH/BPD/31724/2006 from Fundação para a CiĂȘncia e a Tecnologia. Second authorâs research supported by grants MTM 2006-13000-C03-02 from Ministerio de Ciencia y TecnologĂa and CCG 06âUC3M/ESPâ0690 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 TecnologĂa 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
Interlacing properties of zeros of multiple orthogonal polynomials
It is well known that the zeros of orthogonal polynomials interlace. In this
paper we study the case of multiple orthogonal polynomials. We recall known
results and some recursion relations for multiple orthogonal polynomials. Our
main result gives a sufficient condition, based on the coefficients in the
recurrence relations, for the interlacing of the zeros of neighboring multiple
orthogonal polynomials. We give several examples illustrating our result.Comment: 18 page
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