On \(L_p\) extremal polynomials
In this interesting paper, the author considers Lagrange interpolation at the zeros of \(L_p\) extremal polynomials. Let \(\alpha\) be a nondecreasing function on \([-1,1]\) with infinitely many points of increase and assume that \(\int^1_{-1} d\alpha< \infty\). Given \(0<p <\infty\), an \(L_p\) extremal polynomial \(P^*_n (d\alpha,p,x)\) is a monic polynomial of degree \(n\) with \[ \bigl\| P^*_n (d\alpha, p,\cdot) \bigr\|_{d \alpha, p}: =\left(\int^1_{-1} \bigl| P^*_n (d\alpha,p,x) \bigr|^p d\alpha (x)\right)^{1/p} =\inf_{P(x)= x^n+ \dots} \left(\int^1_{-1} \bigl| P(x) \bigr|^p d\alpha(x) \right)^{1/p}. \] Its normalized version is \[ P_n(d\alpha, p,x):= P^*_n (d\alpha, p,x)/ \bigl\| P^*_n (d \alpha, p,\cdot) \bigr\|_{d \alpha,p}. \] In the special case \(p=2\), this is precisely the orthonormal polynomial of degree \(n\) for \(d\alpha\). The author considers Lagrange interpolation at the zeros of \(P_n(d \alpha,p,x)\) and proves the following generalisation of the classical Erdős-Turán theorem: let \(2\leq p<\infty\). Then for every continuous \(f:[-1,1] \to\mathbb{R}\), \[ \lim_{n\to \infty} \int^1_{-1} \bigl| f-L_n(d \alpha,p,f) \bigr|^2 \bigl| P_n (d\alpha, p,\cdot) \bigr|^{p-2} d\alpha=0. \] Here \(L_n(d \alpha,p,f)\) denotes the Lagrange interpolation polynomial of degree \(\leq n\) that interpolates to \(f\) at the zeros of \(P_n(d \alpha,p,x)\). From this he deduces that given \(\varepsilon>0\), there exist sets \(I_n\subset [-1,1]\) with linear measure \(<\varepsilon\) such that for every such \(f\), \[ \lim_{n\to \infty} \int_{[-1,1] \setminus I_n} \bigl| f-L_n(d \alpha,p,f) \bigr|^2d \alpha=0. \] The main tool is a Turán type inequality that bounds integrals of \(| P_n(d \alpha,p,x)|\) below, and earlier general results of the author on weighted Lebesgue function type sums. There are clever ideas both in the formulation of results and the proofs.
scientific article; zbMATH DE number 1143068