Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable

Rakhmanov, Evguenii Andreevich

doi:10.1070/sm1996v187n08abeh000153

Cited by 80 publications

(65 citation statements)

References 2 publications

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“…The equilibrium problem with constraints appeared before mainly in the asymptotic analysis of discrete orthogonal polynomials, see e.g. [6,21,47].…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T . After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

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“…The equilibrium problem with constraints appeared before mainly in the asymptotic analysis of discrete orthogonal polynomials, see e.g. [6,21,47].…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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“…Although these papers do not mention matrix iterations, we can nicely fit our setting in their results. The proof follows along arguments given in [11,21]. We will indicate how we can modify them to the case of para-orthogonal polynomials, who have their zeros on the unit circle.…”

Section: Isometric Arnoldi Minimization Problemmentioning

confidence: 99%

“…Here it will re-appear and we will use the properties and results of §3 with no further comment. [21]. Dragnev and Saff [11] used similar ideas to prove a more general theorem (including external fields), and weakened one of the conditions of Rakhmanov.…”

Section: Isometric Arnoldi Minimization Problemmentioning

confidence: 99%

“…Rakhmanov [21] showed that there exists a unique Borel probability measure µ t that minimizes the logarithmic energy (2.9) among all Borel probability measures µ satisfying 0 tµ σ. He also showed that there exists a constant F t such that (2.10) is satisfied, and that (2.8) and (2.10) characterize the pair (µ t , F t ).…”

Section: Proof Of Theorem 22 Theorem 22 Was Established For Orthogmentioning

confidence: 99%

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Convergence of the Isometric Arnoldi Process

Helsen¹,

Kuijlaars²,

Barel³

2005

SIAM J. Matrix Anal. & Appl.

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It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the rate of convergence and we prove sharpness of these bounds. The distribution of isometric Ritz values is obtained as the minimizer of an extremal problem. We use techniques from logarithmic potential theory in proving these results. Abstract. It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the rate of convergence and we prove sharpness of these bounds. The distribution of isometric Ritz values is obtained as the minimizer of an extremal problem. We use techniques from logarithmic potential theory in proving these results.

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“…Presently, these techniques have been perfected in detail by Gonchar (see [2] and [3]) and Gonchar together with his students (see [4], [5], [6], [7], [8], [9], and [10]) in connection with problems of rational approximation with free poles. Rakhmanov [11] developed this approach for weak asymptotics of polynomials orthogonal with respect to discrete measures by suggesting describing the limit measure of the distribution of zeros of such discrete orthogonal polynomials in the form of the solution of the minimization problem for the energy of the logarithmic potential in the class of measures bounded by the counting measure of the support of the orthogonality measure. Later, this approach was extended in [12] by including an external field.…”

Section: − Q =: ω(γ Q)mentioning

confidence: 99%

Asymptotics of Meixner polynomials and Christoffel–Darboux kernels

Aptekarev

Tulyakov

2013

Trans. Moscow Math. Soc.

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Abstract. We obtain the asymptotics of the classical Meixner polynomials (orthogonal with respect to a discrete measure supported at the nonnegative integer points) and the corresponding reproducing kernels (Christoffel-Darboux kernels) as the number n of the polynomial and the variable x tend to infinity under various relationships between their growth rates. (These asymptotics are known as the Plancherel-Rotach asymptotics.)

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Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable

Cited by 80 publications

References 2 publications

Non-intersecting squared Bessel paths with one positive starting and ending point

Non-intersecting squared Bessel paths with one positive starting and ending point

Convergence of the Isometric Arnoldi Process

Asymptotics of Meixner polynomials and Christoffel–Darboux kernels

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