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Orthogonal polynomials for exponential weights $x^{2\rho} e^{-2Q(x)}$ on [0,$d$). (English) Zbl 1079.42017
This long paper gives a nearly complete treatment of the properties of polynomials orthogonal with respect to exponential weights on (in)finite intervals (bounds, zeros, Christoffel functions etc.). The weights are $$ w(x)=W_{\rho}^2(x)=x^{2\rho}e^{-2Q(x)},\ x\in I=[0,d), $$ with $0<d\leq\infty,\,\rho>-1/2, Q$ continuous and increasing on $I$ with $\lim_{x\uparrow d}\,Q(x)=\infty$. The main results are on -- bounds for the polynomials (Theorems 1.2--1.5 on pages 204/205), -- restricted range inequalities (Theorems 5.1--5.2 on page 220), -- Christoffel functions (Theorems 6.1--6.2 on pages 230/231), -- the zeros (Theorems 7.1--7.2 on page 236). This is a very nicely written paper, entirely within the setting of so-called `hard analysis’.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
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