458 research outputs found
The Analytic Theory of Matrix Orthogonal Polynomials
We give a survey of the analytic theory of matrix orthogonal polynomials.Comment: 85 page
Expanding the Fourier transform of the scaled circular Jacobi ensemble density
The family of circular Jacobi ensembles has a singularity of a type
associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our
interest is in the Fourier transform of the corresponding bulk scaled spectral
density about this singularity, expanded as a series in the Fourier variable.
Various integrability aspects of the circular Jacobi ensemble are used
for this purpose. These include linear differential equations satisfied by the
scaled spectral density for and , and the loop equation
hierarchy. The polynomials in the variable which occur in the
expansion coefficents are found to have special properties analogous to those
known for the structure function of the circular ensemble, specifically
in relation to the zeros lying on the unit circle and interlacing.
Comparison is also made with known results for the expanded Fourier transform
of the density about a guest charge in the two-dimensional one-component
plasma.Comment: 30 page
Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like
integrable systems are connected using the Gauss--Borel factorization of a
Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a
complex quasi-definite measure supported in the unit circle. The factorization
of the moment matrix leads to orthogonal Laurent polynomials in the unit circle
and the corresponding second kind functions. Jacobi operators, 5-term recursion
relations and Christoffel-Darboux kernels, projecting to particular spaces of
truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae
are obtained within this point of view in a completely algebraic way.
Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and
recursion relations, Christoffel-Darboux kernels, projecting to general spaces
of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae
are found in this extended context. Continuous deformations of the moment
matrix are introduced and is shown how they induce a time dependant
orthogonality problem related to a Toda-type integrable system, which is
connected with the well known Toeplitz lattice. Using the classical
integrability theory tools the Lax and Zakharov-Shabat equations are obtained.
The dynamical system associated with the coefficients of the orthogonal Laurent
polynomials is explicitly derived and compared with the classical Toeplitz
lattice dynamical system for the Verblunsky coefficients of Szeg\H{o}
polynomials for a positive measure. Discrete flows are introduced and related
to Darboux transformations. Finally, the representation of the orthogonal
Laurent polynomials (and its second kind functions), using the formalism of
Miwa shifts, in terms of -functions is presented and bilinear equations
are derived
A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character
Maps are polygonal cellular networks on Riemann surfaces. This paper
completes a program of constructing closed form general representations for the
enumerative generating functions associated to maps of fixed but arbitrary
genus. These closed form expressions have a universal character in the sense
that they are independent of the explicit valence distribution of the tiling
polygons. Nevertheless the valence distributions may be recovered from the
closed form generating functions by a remarkable {\it unwinding identity} in
terms of the Appell polynomials generated by Bessel functions. Our treatment,
based on random matrix theory and Riemann-Hilbert problems for orthogonal
polynomials reveals the generating functions to be solutions of nonlinear
conservation laws and their prolongations. This characterization enables one to
gain insights that go beyond more traditional methods that are purely
combinatorial. Universality results are connected to stability results for
characteristic singularities of conservation laws that were studied by
Caflisch, Ercolani, Hou and Landis as well as directly related to universality
results for random matrix spectra as described by Deift, Kriecherbauer,
McLaughlin, Venakides and Zhou
Special functions of Weyl groups and their continuous and discrete orthogonality
Cette thèse s'intéresse à l'étude des propriétés et applications de quatre familles des fonctions spéciales associées aux groupes de Weyl et dénotées , , et . Ces fonctions peuvent être vues comme des généralisations des polynômes de Tchebyshev. Elles sont en lien avec des polynômes orthogonaux à plusieurs variables associés aux algèbres de Lie simples, par exemple les polynômes de Jacobi et de Macdonald. Elles ont plusieurs propriétés remarquables, dont l'orthogonalité continue et discrète. En particulier, il est prouvé dans la présente thèse que les fonctions et caractérisées par certains paramètres sont mutuellement orthogonales par rapport à une mesure discrète. Leur orthogonalité discrète permet de déduire deux types de transformées discrètes analogues aux transformées de Fourier pour chaque algèbre de Lie simple avec racines des longueurs différentes. Comme les polynômes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse numérique. On obtient dans cette thèse quelques formules de >, pour des fonctions de plusieurs variables, en liaison avec les fonctions , et . On fournit également une description complète des transformées en cosinus discrètes de types V--VIII à dimensions en employant les fonctions spéciales associées aux algèbres de Lie simples et , appelées cosinus antisymétriques et symétriques. Enfin, on étudie quatre familles de polynômes orthogonaux à plusieurs variables, analogues aux polynômes de Tchebyshev, introduits en utilisant les cosinus (anti)symétriques.This thesis presents several properties and applications of four families of Weyl group orbit functions called -, -, - and -functions. These functions may be viewed as generalizations of the well-known Chebyshev polynomials. They are related to orthogonal polynomials associated with simple Lie algebras, e.g. the multivariate Jacobi and Macdonald polynomials. They have numerous remarkable properties such as continuous and discrete orthogonality. In particular, it is shown that the - and -functions characterized by certain parameters are mutually orthogonal with respect to a discrete measure. Their discrete orthogonality allows to deduce two types of Fourier-like discrete transforms for each simple Lie algebra with two different lengths of roots. Similarly to the Chebyshev polynomials, these four families of functions have applications in numerical integration. We obtain in this thesis various cubature formulas, for functions of several variables, arising from -, - and -functions. We also provide a~complete description of discrete multivariate cosine transforms of types V--VIII involving the Weyl group orbit functions arising from simple Lie algebras and , called antisymmetric and symmetric cosine functions. Furthermore, we study four families of multivariate Chebyshev-like orthogonal polynomials introduced via (anti)symmetric cosine functions
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