Markov's inequality and zeros of orthogonal polynomials on fractal sets (Q1332288)

A compact subset \(F\) of the space \(\mathbb{R}^m\) is said to preserve Markov's inequality if there exist positive constants \(M\) and \(\alpha\) such that \(\| \text{grad} P\|_\infty \leq Mn^\alpha \| P\|_\infty\) for all polynomials \(P\) in \(m\) variables of degree at most \(n,n=1,2,\dots\). Examples of sets in \(\mathbb{R}^n\) with such a property were discussed in papers by W. Pawłucki and \textit{W. Plésniak} [Math. Ann. 275, No, 3, 467-480 (1986; Zbl 0591.32016); Stud. Math. 88, No. 3, 279-287 (1988; Zbl 0778.26010)]. In another paper by the reviewer [J. Approximation Theory 61, No. 1, 106-117 (1990; Zbl 0702.41023)], it was shown that Markov's property of \(F\) is equivalent to the possibility of Bernstein-type characterization of \({\mathcal C}^\infty\) functions on \(F\) as well as to the existence of a continuous linear operator extending \({\mathcal C}^\infty\) functions from \(F\) to the whole space \(\mathbb{R}^m\). In this paper the author contributes to this theory by showing that in the one-dimensional case the Markov property is closely related to the distribution of the zeros of orthogonal polynomials on \(F\). More precisely, let \(\mu\) be a probability measure supported on \(F\) satisfying for some constants \(c_0>0\) and \(s>0\), \(\mu (B(x,r))\geq c_0r^s\), \(x\in F\), \(0<r\leq 1\). Denote by \(P_n\), \(n=0,1,\dots\), orthogonal polynomials associated to \(\mu\) with the degree of \(P_n=n\). Then, under a density condition for \(F\), the set \(F\) preserves Markov's inequality iff there exist constants \(L > 0\) and \(\beta>0\) such that if \(x_i\) and \(x_{i-1}\) are consecutive zeros of \(P_n\) then \(| x_i-x_{i-1} |\geq L/n^\beta\) for \(n\geq 2\). In particular, the author obtains a distribution rule for the zeros of orthogonal polynomials on some Cantor sets.