Weighted Lebesgue constants: Research problems 2000-1 (Q1968782)

The paper poses the following research problem: Characterize those weight functions \(w:J\to [0,\infty[\), for which there exists a triangular array \(\Gamma=(x_{j,k})_{0\leq j\leq k}\) of interpolation points such that \[ \sup_{x \in J}w(x) \sum^n_{j=1} \bigl|\ell_{j,n}(x) \bigr|w^{-1} (x_{j,n})= O(\log n),\;n\to\infty, \] where \((\ell_{j,n})^n_{j=1}\) are the fundamental polynomials of Lagrange interpolation with \(\ell_{j,n} (x_{k,n}) =\delta_{j,k}\).