Interpolation of entire functions associated with some Freud weights. II (Q1331080)

[For part I see the authors in ibid. 71, No. 2, 123-137 (1992; Zbl 0756.41001).] In this paper, the authors consider geometric rates of convergence of (i) Gauss quadrature, (ii) Lagrange interpolation, (iii) Hermite interpolation when the function \(f\) approximated is entire of suitable order and the interpolation takes place at the zeros of orthogonal polynomials for so-called Freud weights \(e^{-2Q}\). To be more precise, let \(Q: \mathbb{R}\to [0,\infty)\) be even and of smooth polynomial growth at \(\infty\) (by smooth we mean \(Q'''\) exists and satisfies some mild conditions), let \(w\) be a generalized Jacobi factor, so that \(w(x)= \prod^N_{j=1} |z-z_j |^{\delta_j}\) for some complex numbers \(\{z_j\}\) and real exponents \(\{\delta_j\}\), and let \(h: \mathbb{R}\to \mathbb{R}\) be a non-negative measurable function with limit 1 at \(\infty\). Then they consider weights \(W(x):= w(x) h(x) e^{-Q(x)}\) such that \(W^2\) possesses orthonormal polynomials \(p_n (x)\) with \(\int p_n p_m W^2= \delta_{mn}\). The simplest examples are \(W^2 (x)= |x|^\beta \exp(- |x|^\alpha)\), \(x\in \mathbb{R}\). The authors form Gauss-Jacobi quadratures, Lagrange and Hermite interpolation polynomials at the zeros of \(p_n\) for a given entire function \(f\), and guarantee geometric convergence of all these processes if the entire function is of sufficiently slow growth. The exact conditions are formulated in terms of a quantity that is related to the order and type of the entire function.