Mathematics

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product \(\left\langle {f,g} \right\rangle s = \sum\limits_{k - 0}^m {\int\limits_{\Delta _k } {f^{\left( k \right)} \left( x \right)g^{\left( k \right)} \left( x \right)d\mu \kappa } } \left( x \right)\) where \(\left\{ {\mu _\kappa } \right\}_{k = 0}^m ,m \in \mathbb{Z}_ + \) , are measures supported on [−1,1] which satisfy Szegö's condition.

We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f, g) S = f, g + λ f , g where f, g = 1 −1 f (x)g(x)(1 − x 2 ) α−1/2 dx with α > −1/2 and λ > 0. The asymptotics of the zeros and norms of these polynomials is also established.

We consider the orthogonal polynomials on [−1,1] with respect to the weight $$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$$ where h is real analytic and strictly positive on [−1,1] and Ξ c is a step-like function: Ξ c (x)=1 for x∈[−1,0) and Ξ c (x)=c 2, c>0, for x∈[0,1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in ℂ, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n→∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel–Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann–Hilbert problems characterizing the orthogonal polynomials. The local analysis at x=0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

This is a short introduction to the theory of the logarithmic potential in the complex plane. The central ideas are the concepts of energy and equilibrium. We prove some classical results characterizing the equilibrium distribution and discuss the extension of these notions to more general settings when an external field or constraints on the distribution are present. The tools provided by potential theory have a profound impact on different branches of analysis. We illustrate these applications with two examples from approximation theory and complex dynamics.

The concept of k-coherence of two positive measures µ 1 and µ 2 is useful in the study of the Sobolev orthogonal polynomials. If µ 1 or µ 2 are compactly supported on Êthen any 0-coherent pair or symmetrically 1-coherent pair (µ 1 , µ 2 ) must contain a Jacobi measure (up to affine transformation). Here examples of k-coherent pairs (k ≥ 1) when neither µ 1 nor µ 2 are Jacobi are constructed.