Bounds for Lebesgue functions for Freud weights
The role of the Lebesgue function of Lagrange interpolation is well understood for arrays of interpolation points that lie in a fixed finite interval, for example, \([-1,1]\). The size of the Lebesgue function governs the pointwise convergence of the Lagrange interpolation process. It was G. Freud who initiated the study of the Lebesgue function of Lagrange interpolation for arrays of interpolation points over the whole real line. More precisely, he estimated the Lebesgue function when the interpolation points are chosen at the \(n\)th stage to be zeros of the Hermite polynomial of degree \(n\), orthogonal with respect to the weight \(W^ 2_ 2\), where \(W_ 2(x):= \exp(-{1\over 2} x^ 2)\). In this interesting paper, the author extends these results to large classes of Freud weights that include \(W_ \beta(x):= \exp(-{1\over 2}| x|^ \beta)\), \(\beta> 1\). He provides upper and lower bounds that are the same apart from a multiplicative constant. Even for the case of the Hermite weight, these improve on Freud's results. As a sample, if \(\{l_{jn}(x)\}^ n_{j= 1}\) denote the fundamental polynomials of Lagrange interpolation at the zeros \(\{x_{jn}\}^ n_{j= 1}\) of the orthogonal polynomial \(p_ n(x)\) of degree \(n\) with respect to the weight \(W^ 2_ \beta(x)\), then \[ W_ \beta(x) \sum^ n_{k= 1} | l_{kn}(x)|\sim 1+ n^{-1/(2\beta)}| p_ n W_ \beta| (x)\log n= O(\log n) \] uniformly for \(| x|\leq \sigma x_{1n}\). Here \(0< \sigma< 1\) is arbitrary and \(x_{1n}\) denotes the largest zero of \(p_ n\). Moreover, \(\sim\) means that the ratio of the two sides is bounded above and below by positive constants independent of \(n\).
scientific article; zbMATH DE number 703288