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11
General Orthogonal Polynomials
 in “Encyclopedia of Mathematics and its Applications,” 43
, 1992
"... Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. ..."
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Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
Universality Of The Local Eigenvalue Statistics For A Class Of Unitary Invariant Random Matrix Ensembles
, 1997
"... The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Her ..."
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Cited by 54 (4 self)
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The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arose in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions. Key words: random matrices, local asymptotic regime, universality conjecture, orthogonal polynomial technique. 1 Introduction. Problem and results. The random matrix theory (RMT) has been extensively developed and used in a number of areas of theoretical and mathematical physics. In particular the theory provides quite satisfactory description of fluctuations in s...
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields
, 2000
"... The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is pos ..."
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The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the EulerLagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk" in which there is universal behavior involving the sine kernel, and "edge effects" in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any oneparameter family of external fields V=c the equilibrium measure exhibits this regular behavior, except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials and integrable systems.
Extrapolation algorithms and Padé approximations: a historical survey
, 1994
"... This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For ..."
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This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For an extensive bibliography, see [23]. 1 Extrapolation methods Let (S n ) be the sequence to be accelerated. It is assumed to converge to a limit S. An extrapolation method consists in transforming this sequence into a new one, (T n ), by a sequence transformation T : (S n ) \Gamma! (T n ). The transformation T is said to accelerate the convergence of the sequence (S n ) if and only if lim n!1 T n \Gamma S S n \Gamma S =<F13.
Entropy Of Orthogonal Polynomials With Freud Weights And Information Entropies Of The Harmonic Oscillator Potential
, 1998
"... The information entropy of the harmonic oscillator potential V (x) = 1 2 x 2 in both position and momentum spaces can be expressed in terms of the socalled "entropy of Hermite polynomials", i.e. the quantity S n (H) := R +1 \Gamma1 H 2 n (x) log H 2 n (x)e \Gammax 2 dx. These polynomia ..."
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The information entropy of the harmonic oscillator potential V (x) = 1 2 x 2 in both position and momentum spaces can be expressed in terms of the socalled "entropy of Hermite polynomials", i.e. the quantity S n (H) := R +1 \Gamma1 H 2 n (x) log H 2 n (x)e \Gammax 2 dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x) = exp(\Gammajxj m ), m ? 0. Here, firstly the leading term of the entropy of Freud polynomials is found by use of the theory of strong asymptotics of orthogonal polynomials. Then, this result and an improved asymptotical formula of S n (H) are used to analytically study the information entropies of the harmonic oscillator for very excited states (i.e. for large n) in both position and momentum spaces, to be denoted by S ae and S ...
A survey of weighted polynomial approximation with exponential weights
 APPROXIMATION THEORY
, 2007
"... Let W: R! (0, 1] be continuous. Bernstein's approximation problem, posed in 1924,deals with approximation by polynomials in the weighted uniform norm f! kfW kL1(R). Thequalitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem w ..."
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Let W: R! (0, 1] be continuous. Bernstein's approximation problem, posed in 1924,deals with approximation by polynomials in the weighted uniform norm f! kfW kL1(R). Thequalitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. Wesurvey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, MarkovBernstein and Nikolskii inequalities, orthogonal expansionsand Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typicallyeven, and supported on the whole real line, so we exclude Laguerre type weights on [0, 1).Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal
The Support Of The Equilibrium Measure In The Presence Of A Monomial External Field On ...
 1], Trans. Amer. Math. Soc
, 1999
"... The support of the equilibrium measure associated with an external eld of the form Q(x) = cx 2m+1 , x 2 [ 1; 1] with c > 0 and m a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin. 1. ..."
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Cited by 4 (4 self)
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The support of the equilibrium measure associated with an external eld of the form Q(x) = cx 2m+1 , x 2 [ 1; 1] with c > 0 and m a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin. 1.
A problem of Totik on fast decreasing polynomials
, 1998
"... this paper is to solve the following problem posed by V. Totik in [4, Section 13.2, p.202]. ..."
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Cited by 3 (3 self)
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this paper is to solve the following problem posed by V. Totik in [4, Section 13.2, p.202].
Orthogonal Polynomials: from Jacobi to Simon ∗ Contents
, 2005
"... Contents 1 Introduction 3 I General Theory 4 2 Orthogonal polynomials 5 Orthogonal polynomials with respect to measures . . . . . . . . . 5 The RiemannHilbert approach . . . . . . . . . . . . . . . . . . . 7 Orthogonal polynomials with respect to inner products . . . . . . 7 Varying weights . . . ..."
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Contents 1 Introduction 3 I General Theory 4 2 Orthogonal polynomials 5 Orthogonal polynomials with respect to measures . . . . . . . . . 5 The RiemannHilbert approach . . . . . . . . . . . . . . . . . . . 7 Orthogonal polynomials with respect to inner products . . . . . . 7 Varying weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Matrix orthogonal polynomials . . . . . . . . . . . . . . . . . . . 8 3 Classical orthogonal polynomials 8 4 Where do orthogonal polynomials come from? 10 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Pade approximation and rational interpolation . . . . . . . . . . 11 Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Jacobi matrices and spectral theory of selfadjoint operators . . . 13 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Some questions leading to classical orthogonal polynomial