Bounds for Lebesgue Functions for Freud Weights Original Research Article

Let W(x) ≔ e−Q(x), x ∈ R, where Q(x) is even and continuous in R, Q″ is continuous in (0, ∞), and Q′ > 0 in (0, ∞), while for some A, B > 1, [formula]. Let pn(W2, x) denote the nth orthonormal polynomial for the weight W2(x), xkn(W2) the kth zero of pn(W2, x), and lkn(x) the fundamental polynomials. Moreover let an denote the nth Mhaskar-Rahmanov-Saff number for Q and let σ ∈ (0, 1). Then we show that the nth weighted Lebesgue function satisfies uniformly for |x| ≤ σan, [formula]∼ (1 + |x|)−α + √an|Pn(W2, x){(1 + |x|)−αlogn + (1 + |x|)−α̂},≤C{(1 + |x|)−α log n + (1 + |x|)−α̂}, where α ≥ 0 and α̂≔ min{1, α}. We also modify this result to the whole real line.