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Orthogonal polynomials for exponential weights x 2ρ 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 ρ 2 (x)=x 2ρ e -2Q(x) ,xI=[0,d),

with 0<d,ρ>-1/2,Q continuous and increasing on I with lim xd Q(x)=.

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