The Lebesgue Function and Lebesgue Constant of Lagrange Interpolation for Erdoős Weights

AbstractWe 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)∼logn. 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=1logjn)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