Abstract :
We establish pointwise as well as uniform estimates for Lebesgue functions associated with a large class of Erdős weights on the real line. An Erdős weight is of the formW≔exp(−Q), whereQ: R→Ris even and is of faster than polynomial growth at infinity. The archetypal examples areWk, α(x) ≔exp(−Qk, α(x)), ((i))whereQk, α(x) ≔expk(|x|α),α>1,k⩾1. Here expk≔exp(exp(exp(…))) denotes thekth iterated exponential.WA, B(x) ≔exp(−QA, B(x)), ((ii))whereQA, B(x) ≔exp(log(A+x2))B,B>1 andA>A0. For a carefully chosen system of nodesχn≔{ξ1, ξ2,…, ξn},n⩾1, our result imply in particular, that the Lebesgue constant ‖Λn(Wk, α, χn)‖L∞(R)≔supx∈R|Λn(Wk, α, χn)| (x) satisfies uniformly forn⩾N0, ‖Λn(Wk, α, χn)‖L∞(R)∼log n. Moreover, we show that this choice of nodes is optimal with respect to the zeros of the orthonormal polynomials generated byW2. Indeed, letUn≔{xj, n: 1⩽j⩽n},n⩾1, where thexk, nare the zeros of the orthogonal polynomialspn(W2, ·) generated byW2. Then in particular, we have uniformly forn⩾N, ‖Λn(Wk, α, Un)‖L∞(R)∼n1/6(∏kj=1 logj n)1/6. Here,logj≔log(log(log(…))) denotes the j th iterated logarithm. We deduce sharp theorems of uniform convergence of weighted Lagrange interpolation together with rates of convergence. In particular, these results apply toWk, αandWA, B.
