Generalized Christoffel functions for Jacobi-exponential weights on \([-1, 1]\)
The authors study the asymptotic behavior of the \(L^p\)-Christoffel functions \[ \lambda_{p,n}(w;z):=\inf\,\frac{\| Pw\|_{L^p}^p}{| P(z)| ^p}\,, \qquad 0<p<\infty, \] where infimum is taken over all algebraic polynomials \(P\) of degree at most \(n\). The weights \(w\) on \([-1,1]\) are of the form \[ w(x)=e^{-Q(x)}\,\prod_{i=1}^r | x-t_i| ^{p_i}, \qquad t_i\in [-1,1], \quad p_i>\frac1p\,, \quad i=1,2,\ldots,r, \] and \(Q\) belongs to the Lubinsky-Levin class of functions. The main result states that \[ \lambda_{p,n}(w;x)\sim \frac{\sqrt{1-x^2}}n\,e^{-pQ(x)}\,\prod_{i=1}^r \Bigl(| x-t_i| +\frac1n\Bigr)^{p_i} \] uniformly on a certain subset of \([-1,1]\).
scientific article; zbMATH DE number 6699156