##
**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 p_{n}(d \alpha; x) = \gamma_{n}(d \alpha)x^{n}+... (n = 0,1, ...) be the sequence of orthonormal polynomials with respect to the nonnegative measure d \alpha, x_{kn}(d \alpha) (k = 1,2, ... ,n) be the zeros of p_{n}(d \alpha; x) in decreasing order. Let N_{n}(d \alpha; t) be the number of the x_{kn}(d \alpha) satisfying x_{kn}(d \alpha)-x_{nn}(d \alpha) \geq t[x_{1n}(d \alpha)-x_{nn}(d \alpha)]. We say that d \alpha is arc-sine iff **lim**_{n ––> oo}n^{-1}N_{n}(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^{-x2}dx(- oo < x < oo) is arc-sine nor is x^{\rho} e^{-x}dx(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

**lim**_{n ––> oo} \root n-1 \of{\gamma_{n-1}(d \alpha)} [x_{1n}(d \alpha)-x_{nn}(d \alpha)] \geq 4 and that the relation

**lim**_{n ––> oo} \root n-1 \of{\gamma_{n-1}(d \alpha)}[x_{1n}(d \alpha)-x_{nn}(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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag