An update on orthogonal polynomials and weighted approximation on the real line

Spectral Asymptotics of the Non-Self-Adjoint Harmonic Oscillator

“…There is an extensive literature on polynomials which are orthogonal with respect to a non-negative measure µ on the real line; see [5,8,9,11] and references therein. We will need here to consider polynomials p n which are orthogonal with respect to a complex weight σ on R in the following sense.…”

Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

“…One asks for an asymptotic description of p (n) n (z) as n → ∞, which is uniformly valid for all z ∈ C (note that the degree of the polynomial in question coincides with n, the parameter in the measure of orthogonality). The crucial role played by the equilibrium measure in connection with the asymptotics of orthogonal polynomials is well documented (see, for example, [20,28] and the many references contained therein). For example, if x 1 < x 2 < · · · < x n denote the zeros of the n th orthogonal polynomial p n , and if µ n = 1 n ∑ n j=1 δ x j denotes the normalized counting measure for the zeros of p n (here δ x j is the Dirac mass concentrated at x j ), then one has (see [11,23,25,28]) µ n → µ V in the weak- * sense of measures.…”

“…For example, if x 1 < x 2 < · · · < x n denote the zeros of the n th orthogonal polynomial p n , and if µ n = 1 n ∑ n j=1 δ x j denotes the normalized counting measure for the zeros of p n (here δ x j is the Dirac mass concentrated at x j ), then one has (see [11,23,25,28]) µ n → µ V in the weak- * sense of measures. (2.2) As (2.2) indicates, the equilibrium measure is central in more detailed asymptotic descriptions of the orthogonal polynomials [19,20,21,26]. Indeed, if V is real analytic (with sufficient growth), there are six different types of asymptotic formulae that are used to describe the polynomials p n (x) for x ∈ R, catalogued explicitly by the equilibrium measure [6]- [8].…”

Theory of random matrices with strong level confinement: Orthogonal polynomial approach

“…The class of potentials V (x) satisfying all the above requirements is said to be of the Freud type [21]. The typical examples of the Freud potentials are (i)…”