Bounds for weighted Lebesgue functions for exponential weights. II
For sequences of orthogonal polynomials with respect to a weighted inner product, bounds for the weighted Lebesgue functions are provided. Here, the weights are allowed to be exponentials, i.e. \(W(x)=\exp(-Q(x))\) with continuous and convex \(Q\). Thus the Lebesgue function \(\Lambda_n(x)\) is the sum of moduli of the fundamental functions that are divided by the weight function's values at the respective knots, times \(W(x)\). The main theorem gives a short asymptotic expansion of \(\Lambda_n\) for large \(n\). The improvement of this estimate is due to the use of Szabados' idea of adding two extra knots to the sequence.
scientific article; zbMATH DE number 1912071