Endpoint asymptotics for Müntz–Legendre polynomials

Abstract

Muntz–Legendre polynomials Ln(Λ;x) associated with a sequence Λ={λk} are obtained by orthogonalizing the system \((x^{\lambda_{0}},x^{\lambda_{1}},x^{\lambda_{2}},\dots)\) in L2[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is $$\lim_{n\to\infty}L_n\left(1-\frac{y^2}{4S_n}\right)=J_0\big(|y|\big)$$ where \(S_{n}=\sum_{k=0}^{n-1}(2\lambda_{k}+1)+\frac{2\lambda_{n}+1}{2}\) and J0 is the Bessel function of order 0.