Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 294.33006
Autor: Erdös, Paul; Freud, G.
Title: On orthogonal polynomials with regularly distributed zeros. (In English)
Source: Proc. Lond. Math. Soc., III. Ser. 29, 521-537 (1974).
Review: Let pn(d \alpha; x) = \gamman(d \alpha)xn+... (n = 0,1, ...) be the sequence of orthonormal polynomials with respect to the nonnegative measure d \alpha, xkn(d \alpha) (k = 1,2, ... ,n) be the zeros of pn(d \alpha; x) in decreasing order. Let Nn(d \alpha; t) be the number of the xkn(d \alpha) satisfying xkn(d \alpha)-xnn(d \alpha) \geq t[x1n(d \alpha)-xnn(d \alpha)]. We say that d \alpha is arc-sine iff limn ––> oon-1Nn(d \alpha; t) = 1/2 -{1 \over \pi} \arcsin(2t-1). [J. L. Ullman, Proc. London math. soc., III. Ser. 24, 119-148 (1972; Zbl 232.33007)]. By well-known properties of the zeros of the classical orthogonal polynomials, (1-x)\beta(1+x)\gamma dx(-1 < x < 1) is arc-sine for \beta , \gamma > -1 but neither e-x2dx(- oo < x < oo) is arc-sine nor is x\rho e-xdx(0 < x < oo) arc-sine for any \rho > -1.
A class of absolutely continuous arc-sine measures with non-compact support was discovered by P. Erdös [Proc. Conf. construct. Theory Functions (Approximation Theory) 1969, 145-150 (1972; Zbl 234.33014)]. The authors prove that we have for arbitrary d \alpha
limn ––> oo \root n-1 \of{\gamman-1(d \alpha)} [x1n(d \alpha)-xnn(d \alpha)] \geq 4 and that the relation
limn ––> oo \root n-1 \of{\gamman-1(d \alpha)}[x1n(d \alpha)-xnn(d \alpha)] = 4 (*) implies that d \alpha is arc-sine. (*) is not only sufficient but also necessary if d \alpha = wdx is absolutely continuous and either it has compact support or w(x) = \exp {-2Q(|x|)} (-oo < x < oo) where Q(x) (x \geq 0) is a positive increasing differentiable function for which x\rho Q'(x) is increasing for some \rho < 1. An example is constructed of an absolutely continuous arc-sine measure d \alpha for which (*) does not hold.
Following J.L.Ullman, loc. cit. we say that A \subset [-1,1] is a determining set if every absolutely continuous d \alpha = w(x)dx which satisfies A \subseteq {x: w(x) > 0 } \subseteq [-1,1] is arc-sine on [-1,1]. We give a proof of the conjecture of P.Erdös that a measurable set A is a determining set if and only if it has minimal logarithmic capacity 1/2 .
Classif.: * 33A65 33A65
42C05 General theory of orthogonal functions and polynomials
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