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38
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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Cited by 59 (6 self)
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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.
The RiemannHilbert approach to strong asymptotics for orthogonal polynomials on [1, 1]
"... We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [ ..."
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Cited by 44 (23 self)
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We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [1, 1], for the recurrence coe#cients and for the leading coe#cients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for RiemannHilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szego function associated with the weight and for the local analysis around the endpoints 1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. 1 Supported by FWO research project G.0176.02 and by INTAS project 00272 2 Supported by NSF grant #DMS9970328 3 Supported by FWO research project G.0184.01 and by INTAS project 00272 4 Research Assistant of the Fund for Scientific Research  Flanders (Belgium) 1 1
Hilbert analysis for orthogonal polynomials
 Orthogonal Polynomials and Special Functions (E. Koelink and W. Van Assche eds.) Lecture Notes in Mathematics 1817 (2003
"... Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on ..."
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Cited by 16 (8 self)
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Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on [−1, 1] where α, β> −1 and h is real analytic and positive on [−1, 1]. These notes are based on joint work with
Universality of sinekernel for Wigner matrices with a small Gaussian perturbation
, 2009
"... We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N −3/4+β for some positive β> 0. We prove that the local eigenvalue statistics follows the universal ..."
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Cited by 14 (8 self)
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We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N −3/4+β for some positive β> 0. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.
Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials
 JOURNAL OF APPROXIMATION THEORY
, 2007
"... We apply universality limits to asymptotics of spacing of zeros fxkng of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n!1 xkn xk+1;n ~ Kn (xkn; xkn) = 1 under minimal hypotheses on the weight, with ~ Kn denoting a normalized reproduc ..."
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Cited by 12 (4 self)
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We apply universality limits to asymptotics of spacing of zeros fxkng of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n!1 xkn xk+1;n ~ Kn (xkn; xkn) = 1 under minimal hypotheses on the weight, with ~ Kn denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the di¤erentiated kernels K (r;s) nX 1 n (x; x) = k=0 p (r) k (x) p(s) k (x):
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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Cited by 7 (1 self)
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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
Complex Gaussian quadrature of oscillatory integrals
, 2008
"... We construct and analyze Gausstype quadrature rules with complexvalued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to t ..."
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Cited by 6 (5 self)
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We construct and analyze Gausstype quadrature rules with complexvalued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with nonstandard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.
DECOMPOSITION OF SPACES OF DISTRIBUTIONS INDUCED BY HERMITE EXPANSIONS
, 705
"... Abstract. Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the TriebelLizorkin and Besov spaces on R d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown t ..."
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Cited by 4 (4 self)
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Abstract. Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the TriebelLizorkin and Besov spaces on R d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite TriebelLizorkin and Besov spaces are, in general, different from the respective classical spaces. 1.
EXPLICIT ORTHOGONAL POLYNOMIALS FOR RECIPROCAL POLYNOMIAL WEIGHTS ON ( 1; 1)
"... Abstract. Let S be a polynomial of degree 2n + 2, that is positive on the real axis, and let w = 1=S on ( 1; 1). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical BernsteinSzego formula fo ..."
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Cited by 3 (3 self)
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Abstract. Let S be a polynomial of degree 2n + 2, that is positive on the real axis, and let w = 1=S on ( 1; 1). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical BernsteinSzego formula for ( 1; 1). Orthogonal Polynomials, BernsteinSzego formulas. 42C05 1. The Result 1 The BernsteinSzego formula provides an explicit formula for orthogonal polynomials for a weight of the form p 1 x 2 =S (x) ; x 2 ( 1; 1) ; where S is a polynomial positive in ( 1; 1), possibly with at most simple zeros at 1. It plays a key role in asymptotic analysis of orthogonal polynomials. In this paper, we present an explicit formula for the nth degree orthogonal polynomial for weights w on the whole real line of the form (1.1) w = 1=S; where S is a polynomial of degree 2n + 2, positive on R. In addition, we give representations for the (n + 1)st reproducing kernel and Christo¤el function. We present elementary proofs, although they follow partly from the theory of de Branges spaces [1]. The formulae do not seem to be recorded in de Branges’book, nor in the orthogonal polynomial literature [2], [3], [7], [8], [9]. We believe they will be useful in analyzing orthogonal polynomials for weights on R. Recall that we may de…ne orthonormal polynomials fpmg n m=0, where (1.2) pm (x) = mx m +:::, m> 0; satisfying Z 1 pjpkw = 1 Because the denominator S in w has degree 2n + 2, orthogonal polynomials of degree higher than n are not de…ned. The (n + 1) st reproducing kernel for w is nX (1.3) Kn+1 (x; y) = pj (x) pj (y): j=0 Inasmuch as S is a positive polynomial, we can write jk: