49 research outputs found
Asymptotic zero distribution of orthogonal polynomials with discontinously varying recurrence coefficients
No full textThe zero distribution of orthogonal polynomials pn, N, n = 0, 1, ⋯ generated by recurrence coefficients an, N and bn, N depending on a parameter N has been recently considered by Kuijlaars and Van Assche under the assumption that an, N and bn, N behave like a(n/N) and b(n/N), respectively, where a(·) and b(·) are continuous functions. Here, we extend this result by allowing a(·) and b(·) to be measurable functions so that the presence of possible jumps is included. The novelty is also in the sense of the mathematical tools since, instead of applying complex analysis arguments, we use recently developed results from asymptotic matrix theory due to Tyrtyshnikov, Serra Capizzano, and Tilli
Weighted Approximation with Varying Weights: The Case of a Power-Type Singularity
No full textAbstractThe class of functions that can be uniformly approximated by weighted polynomials of the formwnPnwith degPn≤n, depends on the behavior of the extremal measure associated withwas introduced by Mhaskar and Saff. It is shown that if in a neighborhood of a pointt0the extremal measure has a density with a power-type singularity att0, then every uniform limit vanishes att0. This complements results of Totik for continuous positive densities and Kuijlaars for densities that vanish att0
Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature
No full text33 pages, 11 figures.-- Pre-print article.In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral on the real axis with a high order stationary point, and their limit distribution is also analyzed. We show that the zeros accumulate along a contour in the complex plane that has the S-property in an external field. In addition, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann-Hilbert problem.A. Deaño acknowledges financial support from the programme of postdoctoral grants of the Spanish Ministry of Education and Science and project MTM2006-09050. D. Huybrechs is a Postdoctoral Fellow of the Research Foundation Flanders (FWO) and is supported by FWO-Flanders project G061710N. A.B.J. Kuijlaars is supported by K.U. Leuven research grant OT/08/33, FWO-Flanders project G.0427.09, by the Belgian Interuniversity
Attraction Pole P06/02, by the European Science Foundation Program MISGAM, and by grant MTM2008-06689-C02-01 of the Spanish Ministry of Science and Innovation.No publicad
A Christoffel–Darboux formula for multiple orthogonal polynomials
No full textAbstractBleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel–Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Riemann–Hilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis
Chebyshev-type quadrature for analytic weights on the circle and the interval
No full textAbstractWe give a sharp asymptotic bound on the number of nodes needed for Chebyshev-type (= equal weight) quadrature of degree p for measures on [−1, 1] of the form w(t)(π√ 1 − t2)dt, where w is positive on [−1, 1] and analytic in a neighborhood of [−1, 1]. This bound is derived from a corresponding bound for Chebyshev-type quadrature for analytic weights on the unit circle. In addition, we present some results on Chebyshev-type quadrature on certain algebraic curves
Equivalence of QCD in the epsilon-regime and chiral Random Matrix Theory with or without chemical potential
No full textWe prove that QCD in the epsilon-regime of chiral Perturbation Theory is equivalent to chiral Random Matrix Theory for zero and both non-zero real and imaginary chemical potential mu. To this aim we prove a theorem that relates integrals over fermionic and bosonic variables to super-Hermitian or super-Unitary groups also called superbosonization. Our findings extend previous results for the equivalence of the partition functions, spectral densities and the quenched two-point densities. We can show that all k-point density correlation functions agree in both theories for an arbitrary number of quark flavors, for either mu=0 or mu=/=0 taking real or imaginary values. This implies the equivalence for all individual k-th eigenvalue distributions which are particularly useful to determine low energy constants from Lattice QCD with chiral fermions
A Note on Weighted Polynomial Approximation with Varying Weights
No full textAbstractIt is shown that if weighted polynomialswnPnwith degPn⩽nconverge uniformly on the support of the extremal measure associated withw, then they converge to 0 everywhere else. It is also shown that uniform approximation on the support can always be characterized by a closed subsetZhaving the property that a function can be approximated if and only if it vanishes onZ
A vector equilibrium problem for Muttalib-Borodin biorthogonal ensembles
No full textThe Muttalib-Borodin biorthogonal ensemble is a joint density function for n particles on the positive real line that depends on a parameter θ. There is an equilibrium problem that describes the large n behavior. We show that for rational values of θ there is an equivalent vector equilibrium problem.sponsorship: Long term structural funding -Methusalem grant of the Flemish Governmentstatus: Publishe
