A note on strong asymptotics of weighted Chebyshev polynomials (Q2441940)

The author considers the Chebyshev polynomials normalized with respect to a positive continuous weight function \(w\) on \([-1,1]\), denoted by \(T_n(x,w)\), and studies their asymptotic behavior for large degree \(n\). This analysis was initiated in a previous paper of the author and \textit{F. Peherstorfer} [Math. Proc. Camb. Philos. Soc. 144, No. 1, 241--254 (2008; Zbl 1160.33007)], where the authors show that, when the weight \(w\) is a \(C^{2+\alpha}\) function, \(\alpha>0\), then \(T_n(\cos\phi,w)=\Re\{ e^{-in\phi}\pi^2(e^{i\phi})\}+o(1)\) as \(n\to\infty\) uniformly for \(\phi\in[0,\pi]\), where \(\pi(z)\) is the Szegö function corresponding to the weight \(w\). In this paper, the author extends the range of validity of this asymptotic formula and shows that it is true in a larger space of weight functions, namely, the space \(C^{1+}\) of functions whose first derivatives are of the Dini-Lipschitz class, relaxing in this way the smoothness requirement for the weight \(w\). The key point in the proof of this result is a sharper bound for the \(n\)th weighted strong unicity constant \(\gamma_n(f,w)\) of a continuous function \(f\) in \([-1,1]\) relative to the weight \(w\). Whereas in [\textit{A. Kroó} and \textit{F. Peherstorfer}, Math. Proc. Camb. Philos. Soc. 144, No. 1, 241--254 (2008; Zbl 1160.33007)], it is shown that \(\gamma_n(T_n(.,1/\rho_m),1/\rho_m)={\mathcal O}(n^2)\) as \(n\to\infty\), with \(\rho_m\) a polynomial of degree \(m<n\) and positive on \([-1,1]\), in this paper it is shown that \(\gamma_n(T_n(.,1/\rho_m),1/\rho_m)={\mathcal O}(n)\) as \(n\to\infty\). Moreover, it is shown that this asymptotic estimate is sharp in general. The fact that the estimate \(\gamma_n(T_n(.,1/\rho_m),1/\rho_m)={\mathcal O}(n)\) as \(n\to\infty\) is sharp in general motivates the author to set the following conjecture: there exist positive Lip1 weights \(w\) on \([-1,1]\) non-differentiable at some point of \((-1,1)\) for which the asymptotic relation \(T_n(\cos\phi,w)=\Re\{ e^{-in\phi}\pi^2(e^{i\phi})\}+o(1)\) as \(n\to\infty\) fails.