A. B. J. Kuijlaars

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Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

A. B. J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky

Communications in Mathematical Physics > 2011 > 308 > 1 > 227-279

We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t ...

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Random matrix model with external source and a constrained vector equilibrium problem

P. Bleher, S. Delvaux, A. B. J. Kuijlaars

Communications on Pure and Applied Mathematics > 64 > 1 > 116 - 160

We consider the random matrix model with external source in the case where the potential V(x) is an even polynomial and the external source has two eigenvalues ±a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures (μ₁, μ₂) that solve a constrained vector equilibrium problem. The proof is based...

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Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

A. B. J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky

Communications in Mathematical Physics > 2009 > 286 > 1 > 217-275

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard...

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Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

A. B. J. Kuijlaars, W. Assche, F. Wielonsky

Constructive Approximation > 2005 > 21 > 3 > 351-412

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Padé approximation to the exponential function, defined by p(z)e^-z + q(z) + r(z) e^z = O(z³ⁿ⁺²) as z → 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform...

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Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

A. B. J. Kuijlaars, K. T-R McLaughlin

Constructive Approximation > 2004 > 20 > 4 > 497-523

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\!\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the...

article

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

A. B. J. Kuijlaars, K. T-R McLaughlin

Constructive Approximation > 2003 > 20 > 4

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\!\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit...

article

On The Sharpness of an Asymptotic Error Estimate for Conjugate Gradients

B. Beckermann, A. B. J. Kuijlaars

BIT Numerical Mathematics > 2001 > 41 > 5 > 856-867

Recently, the authors obtained an upper bound on the error for the conjugate gradient method, which is valid in an asymptotic setting as the size of the linear systems tends to infinity. The estimate depends on the asymptotic distribution of eigenvalues, and the ratio between the size and the number of iterations. Such error bounds are related to the existence of polynomials with value 1 at 0 whose...

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Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

A. B. J. Kuijlaars

Constructive Approximation > 2000 > 16 > 4

Abstract. We consider the asymptotic zero behavior of polynomials that are extremal with respect to slowly decaying weights on [0, fty) , such as the log-normal weight \exp(-2log2 x) . The zeros are contracted by taking the appropriate dn th roots with dnfty . The limiting distribution of the contracted zeros is described in terms of the solution of an extremal problem in logarithmic potential theory...

article

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

A. B. J. Kuijlaars

Constructive Approximation > 2000 > 16 > 4 > 559-589

. We consider the asymptotic zero behavior of polynomials that are extremal with respect to slowly decaying weights on [0, ∈fty) , such as the log-normal weight \exp(-γ ²log ² x) . The zeros are contracted by taking the appropriate d _n th roots with d _n →∈fty . The limiting distribution of the contracted zeros is described in terms of the solution of an extremal problem in logarithmic potential...

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The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

A. B. J. Kuijlaars, W. Van Assche

Journal of Approximation Theory > 1999 > 99 > 1 > 167-197

We study the zeros of orthogonal polynomials p _n _, _N , n=0, 1, ..., that are generated by recurrence coefficients a _n _, _N and b _n _, _N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials p _n _, ...

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A Problem of Totik on Fast Decreasing Polynomials

A. B. J. Kuijlaars, W. Van Assche

Constructive Approximation > 1998 > 14 > 1 > 97-112

. We solve a problem posed by V. Totik on the existence of fast-decreasing polynomials p _n of degree $\leq n$ with p _n (0)=1 and $|p_n(x)|\leq C\exp(-cn|x|^\beta)$ for $x \in [-1,1]$ . For $0 \leq \beta \leq 2$ the largest c for which such polynomials exist was known. We give the solution for β > 2 .

article

A Problem of Totik on Fast Decreasing Polynomials

A. B. J. Kuijlaars, W. Van Assche

Constructive Approximation > 1998 > 14 > 1 > 97-112

Abstract. We solve a problem posed by V. Totik on the existence of fast-decreasing polynomials pn of degree $\leq n$ with pn(0)=1 and $|p_n(x)|\leq C\exp(-cn|x|^\beta)$ for $x \in [-1,1]$ . For $0 \leq \beta \leq 2$ the largest c for which such polynomials exist was known. We give the solution for 2 .

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Search results for: A. B. J. Kuijlaars

Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

Random matrix model with external source and a constrained vector equilibrium problem

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

On The Sharpness of an Asymptotic Error Estimate for Conjugate Gradients

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

A Problem of Totik on Fast Decreasing Polynomials

A Problem of Totik on Fast Decreasing Polynomials

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